Research Article Second-Order Systems of ODEs Admitting ...downloads.hindawi.com/journals/jam/2013/147921.pdf · We present illustrative examples of familiar integrable physical systems

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 147921 15 pageshttpdxdoiorg1011552013147921

Research ArticleSecond-Order Systems of ODEs Admitting Three-DimensionalLie Algebras and Integrability

Muhammad Ayub12 Masood Khan1 and F M Mahomed2

1 Department of Mathematics Quaid-i-Azam University Islamabad 45320 Pakistan2 Centre for Differential Equations Continuum Mechanics and Applications School of Computational and Applied MathematicsUniversity of the Witwatersrand Wits 2050 South Africa

Correspondence should be addressed to F M Mahomed fazalmahomedwitsacza

Received 27 November 2012 Accepted 20 January 2013

Academic Editor Mehmet Pakdemirli

Copyright copy 2013 Muhammad Ayub et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We present a systematic procedure for the determination of a complete set of kth-order (119896 ge 2) differential invariants correspondingto vector fields in three variables for three-dimensional Lie algebras In addition we give a procedure for the construction of a systemof two kth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that thereare 29 classes for the case of k = 2 and 31 classes for the case of 119896 ge 3 We discuss the singular invariant representations of canonicalforms for systems of two second-order ODEs admitting three-dimensional Lie algebras Furthermore we give an integrationprocedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprisesof two approaches namely division into four types I II III and IV and that of integrability of the invariant representations Weprove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra then its general solution can beobtained from a partially linear partially coupled or reduced invariantly represented system of equations A natural extension ofthis result is provided for a system of two kth-order (119896 ge 3) ODEs We present illustrative examples of familiar integrable physicalsystems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systemsthat give rise to closed trajectories

1 Introduction

Realizations of Lie algebras in terms of vector fields areapplied in particular for the integration and classificationof ordinary differential equations (see eg [1ndash5]) In spiteof the importance for applications the problem of completedescription of realizations has not been solved even forcases when either the dimension of the algebras or thedimension of the realization space is a fixed small integer[6] The classification of finite-dimensional Lie algebras wasinitiated by Lie [7] who also presented realizations in termsof vector fields in (1+1)- and (1+2)-dimensional spaces Inthe case of vector fields in the two-dimensional space hegave a complete classification in the complex domain Forrealizations of vectors fields in the three-dimensional spaceLie presented a partial classification which was recently com-pleted All the possible complex Lie algebras of dimension

less than four were already obtained by Lie himself [8 9]Bianchi [10] investigated three-dimensional real Lie algebrasComplete and correct classification of four-dimensional realLie algebras was obtained by Mubarakzjanov [11] WafoSoh and Mahomed [12] used the results of Bianchi andMubarakzjanov to classify realizations of three- and four-dimensional real Lie algebras in the space of three variablesand they also described systems of two second-order ODEsadmitting real four-dimensional Lie algebras

The realizations of real Lie algebras of dimension nogreater than four of vector fields in the space of an arbitrary(finite) number of variables were given in [6] Popovych etal [6] used the results of Bianchi and Mubarakzjanov with adifferent approach They found that the work [12] does notprovide the complete list of realizations of real Lie algebras ofdimensions three and four

2 Journal of Applied Mathematics

Systems of two second-order ODEs arise frequently inseveral applications in classical fluid and quantum mechan-ics as well as in general relativity Hence these have beenstudied vigorously over the years A sample of the literature asfar as symmetry Lie algebras of ODEs are concerned can befound in [13ndash15] In particular the classification of systemsof two second-order ODEs admitting real four-dimensionalLie algebras was attempted in [12] Gaponova andNesterenko[16] gave a classification of system of two second-order ODEsadmitting real three- and four-dimensional Lie algebras byusing the results of [6]

In the this paper by utilizing realizations given in [6]a systematic procedure for finding a complete set of 119896th-order (119896 ge 2) differential invariants including basis ofinvariants corresponding to vector fields in three variablesfor three-dimensional Lie algebras is presented In additionwe give a procedure for the construction of two 119896th-orderODEs possessing three-dimensional Lie algebras from theassociated complete set of invariants We show that thereare 29 classes for 119896 = 2 and 31 classes for 119896 ge 3Moreover we mention those cases in which regular systemsof two second-order ODEs are not obtained for three-dimensional Lie algebras We present a brief discussion onthe singular representation of canonical forms for systemsof two second-order ODEs admitting three-dimensional Liealgebras Furthermore we provide an integration procedurefor canonical forms for systems of two second-order ODEsadmitting three-dimensional Lie algebras This procedure iscomposed of two approaches namely one which depends ongeneral observations of canonical forms which admit three-dimensional Lie algebras the second depends on a completeset of differential invariants and is applicable only for thosecases which admit three-dimensional solvable Lie algebrasWe show by familiar physical examples how these integrationapproaches work In the second approach for solvable Liealgebras we analyze integrability in terms of integration ofthe invariant representations We prove that if a system oftwo second-order ODEs admits a solvable three-dimensionalLie algebra then its general solution can be deduced from apartially linear partially coupled or invariant reduced systemof equations A natural extension of this result is stated for asystem of two 119896th-order (119896 ge 3) ODEs

In Table 1 for ease of reference we list the realizations ofthree-dimensional Lie algebras as given in [6] In the secondsection for the Lie algebras of vector fields considered wediscuss and deduce a complete set of differential invariantsincluding basis of invariants with the construction of thecorresponding systems of two 119896th-order ODEs Moreoverfor systems of two second-order ODEs admitting three-dimensional Lie algebras we discuss the singular represen-tation and those cases in which we do not obtain systemsIn the end in Table 2 we give a complete list of second-order differential invariants corresponding to vector fields inthree variables of three-dimensional real Lie algebras and theassociated complete list of canonical forms for systems of twosecond-order ODEs In Section 3 we provide an integrationprocedure for the canonical forms of systems of two 119896th-order(119896 ge 2) ODEs admitting three-dimensional Lie algebrasThisis stated as theorems for systems admitting solvable algebras

Reduction of canonical forms for systems of two second-orderODEs into invariant systems of first-order and algebraicequations are given in Table 2 The paper ends with a briefconclusion

2 Invariants and Systems of ODEs

When one calculates the symmetries of a given differentialequation one finds the generators in the form of vectorfields and then computes the Lie brackets to get the structureconstants of the particular Lie algebra one has found We canstart from a given Lie algebra with a set of structure constantsand ask which vector fields in at most three variables satisfythe given Lie bracket relations with none of the vectorfields vanishing We thus ask for possible realizations ofthe given Lie algebra We make use of the realizationsof three-dimensional Lie algebras in the (1+2)-dimensionalspace given in [6] We utilize these realizations in order toobtain differential invariants and invariant representationsfor system of two second-order and higher-order ODEsExtensions to higher order invariants are given to understandthe behavior of such invariantsThis is the inverse problem insymmetry analysis and was initiated by Lie

For completeness and ease of reference we list therealizations of three-dimensional real algebras as given in [6]in Table 1 However we mostly keep the format of [12]

Differential invariants play an important role in thestructure and reduction of differential equations Differen-tial invariants can be obtained by various approaches andwe refer for example to [16] for realizations in (1+2)-dimensional space Differential invariants obtained by var-ious approaches corresponding to three-dimensional Liealgebras have very interesting properties In the investigationof the invariant approach for a system of two second-orderODEs admitting three-dimensional Lie algebras it is foundthat for a regular system ofODEs there are four independentinvariants three of which play the role of basis and the fourthone is derived from the remaining three

For the case of a regular system of two second-orderODEs admitting three-dimensional Lie algebras there arefour functionally independent invariants For example con-sider the system of two second-order ODEs

2

= 119891(119910

)

2minus

3

= 119892(119910

) (1)

admitting the three-dimensional Lie algebra 1198603

32 We have

the four independent invariants

1199081= 119910 119908

2=

119908

3=

2 119908

4=

2minus

3

(2)

Here the invariants1199083and 119908

4are of order two that is equal

to the order of the system of ODEs that is constructible

Journal of Applied Mathematics 3

Table 1 Realizations of three-dimensional Lie algebras in three variables

Algebra N Realization Rank

11986031

1 120597119905 120597119909 120597119910

3

2 120597119909 120597119910 119905120597119909+ 119891(119905)120597

119910 2

3 120597119909 119905120597119909 119910120597119909

1

4 120597119905 119909120597119905 120601(119909)120597

119905 12060110158401015840(119909) = 0 1

11986032

[1198831 1198832] = 119883

1

1 120597119905 119905120597119905+ 120597119910 120597119909

3

2 120597119905 119905120597119905+ 119910120597119909 120597119909

2

3 120597119905 119905120597119905 120597119909

2

4 120597119909 119905120597119905+ 119909120597119909 119905120597119909

2

11986033

[1198832 1198833] = 119883

1

1 120597119905 120597119909 119909120597119905

2

2 120597119905 120597119909 119909120597119905+ 119910120597119909

2

3 120597119905 120597119909 119909120597119905+ 120597119910

3

11986034

1 120597119905 120597119909 (119905 + 119909)120597

119905+ 119909120597119909

2

[1198831 1198833] = 119883

12 120597

119905 120597119909 (119905 + 119909)120597

119905+ 119909120597119909+ 120597119910

3

[1198832 1198833] = 119883

1+ 1198832

3 120597119909 119905120597119909 minus120597119905+ 119909120597119909

2

11986035

[1198831 1198833] = 119883

1

[1198832 1198833] = 119883

2

1 120597119905 120597119909 119905120597119905+ 119909120597119909

2

2 120597119905 120597119909 119905120597119905+ 119909120597119909+ 120597119910

3

3 120597119909 119905120597119909 119909120597119909

1

4 120597119909 119905120597119909 119909120597119909+ 120597119910

2

119860119886

36 |119886| le 1 119886 = 0 1 1 120597

119905 120597119909 119905120597119905+ 119886119909120597

1199092

[1198831 1198833] = 119883

12 120597

119905 120597119909 119905120597119905+ 119886119909120597

119909+ 120597119910

3

[1198832 1198833] = 119886119883

23 120597

119909 119905120597119909 (1 minus 119886)119905120597

119905+ 119909120597119909

2

119860119886

37 119886 ge 0 1 120597

119905 120597119909 (119886119905 + 119909)120597

119905+ (119886119909 minus 119905)120597

1199092

[1198831 1198833] = 119886119883

1minus 1198832

2 120597119905 120597119909 (119886119905 + 119909)120597

119905+ (119886119909 minus 119905)120597

119909+ 120597119910

3

[1198832 1198833] = 119883

1+ 1198861198832

3 120597119909 119905120597119909 minus(1 + 119905

2)120597119905+ (119886 minus 119905)119909120597

1199092

11986038

sl(2 119877)

[1198831 1198832] = 119883

1

[1198832 1198833] = 119883

3

[1198833 1198831] = minus2119883

2

1 120597119905 119905120597119905 1199052120597119905

1

2 120597119905 119905120597119905+ 119909120597119909 1199052120597119905+ 2119905119909120597

119909+ 119909120597119910

3

3 120597119905+ 120597119909 119905120597119905+ 119909120597119909 1199052120597119905+ 1199092120597119909

2

4 minus1199051205971199091

2(minus119905120597119905+ 119909120597119909) 119909120597119905

2

5 120597119905 119905120597119905+ 119909120597119909 (1199052minus 1199092)120597119905+ 2119905119909120597

1199092

11986039

so(3)[1198831 1198832] = 119883

3

1 (1 + 1199052)120597119905+ 119909119905120597

119909 119909120597119905minus 119905120597119909 minus119909119905120597

119905minus (1 + 119909

2)120597119909

2

[1198833 1198831] = 119883

2

[1198832 1198833] = 119883

1

2 minus sin 119905 tan119909120597119905minus cos 119905120597

119909+ sin 119905 sec119909120597

119910 minus cos 119905 tan119909120597

119905+ sin 119905120597

119909+ cos 119905 sec119909120597

119910 120597119905

3

Similarly for the regular system of two third-order ODEs119909

2

minus22

3

= 119891(119910

2

2minus

3)

minus119910

4

+

1199092minus 3 + 3

2

5

= 119892(119910

2

2minus

3)

(3)

admitting the sameLie algebra119860332 we have the six invariants

1199081= 119910 119908

2=

119908

3=

2 119908

4=

2minus

3

1199085=

119909

2

minus22

3 119908

6= minus

119910

4

+

1199092minus 3 + 3

2

5

(4)

from which we determine the third-order system Here theinvariants 119908

5and 119908

6are of order three that is equal to

the order of the considered system of ODEs By proceedingin this way for regular systems of two 119896th-order ODEsadmitting three-dimensional Lie algebras the number oforder 119896 invariants is two

To study differential invariants further we consider thefollowing example Consider the system

= 119891 (119890

119910 119886minus1

)

119890minus119910

119886minus1

2+

+(119886 minus 1) 119890

minus119910119886minus2

2+

= 119892 (119890119910 119886minus1

)

(5)


Table 2 Second-order differential invariants and equations

Algebra N Invariants and equations

11986031

1 119906 = V = 119908 = 119889V

119889119906=

= 119891(119906 V)

= 119892(119906 V)

2 119906 = 119905 V = minus 1198911015840(119905) 119908 =

119889V

119889119906= minus 119891

1015840(119905) minus 119891

10158401015840(119905)

= ℎ(119906 V) minus 1198911015840(119905) minus 119891

10158401015840(119905) = 119896(119906 V)

3 119906 = 119905 V = 119910 119908 = 119889V

119889119906= and = 0

lowast = 119891(119906 V119889V

119889119906) = 0

11986032

1 119906 = 119890119910 V =

119908 =

2

119889V

119889119906=

119890119910(2+ )

minus

1198901199102(2+ )

2

= 119891(119906 V)

119890119910(2+ )

minus

1198901199102(2+ )

= 119892(119906 V)

2 119906 = 119910 V = 119890

119908 =

2

119889V

119889119906= 119890

+ 119890

( minus )

2

2

= 119891(119906 V) 119890

+ 119890

( minus )

2

= 119892(119906 V)

3 119906 = 119910 V =

119908 =

2

119889V

119889119906=

2minus

3

2

= 119891(119906 V)

2minus

3

= 119892(119906 V)

4 119906 = 119910 V = 119905 119908 = 119905119889V

119889119906= 1 +

119905

119905 = 119891(119906 V) 1 +119905

= 119892(119906 V)

11986033

1 119906 = 119910 V =

119908 =

3

119889V

119889119906=

2minus

3

3

= 119891(119906 V)

2minus

3

= 119892(119906 V)

2 119906 = 119910 V =2

22minus

1

119908 =

3

minus

2

119889V

119889119906=

3

+( minus

2)

4

3

minus

2

= 119891(119906 V) 3

+( minus

2)

4

= 119892(119906 V)

3 119906 = 119910 minus1

V =

119908 =

3

119889V

119889119906=

2

( + 2)minus

3

2( +

2)

3

= 119891(119906 V) 2

( + 2)minus

3

2( +

2)= 119892(119906 V)

11986034

1 119906 = 119910 V = 1198901

119908 =

2

119889V

119889119906= 1198901

minus 1198901

(1 + )

3

2

= 119891 (119906 V) 1198901

minus 1198901

(1 + )

3

= 119892(119906 V)

2 119906 = 119910minus1

V = 119890

1

119908 =

2

119889V

119889119906= 1198901

+ 2

minus1198901

(1 + )

( + 2)

2

= 119891(119906 V) 1198901

+ 2

minus 1198901

(1 + )

( + 2)

= 119892(119906 V)

3 119906 = 119910 V = 119908 = 119890119905

119889V

119889119906=

119890119905 = 119891 (119906 V)

= 119892(119906 V)

11986035

1 119906 = 119910 V = 119908 =

2

119889V

119889119906=

2

= 119891(119906 V)

= 119892(119906 V)


Table 2 Continued


2 119906 = V = 119890119910 119908 =

119889V

119889119906= 1198901199102

+ 119890119910

= 119891(119906 V) 119890119910

2

+ 119890119910

= 119892(119906 V)

3 119906 = 119905 V = 119910119889V

119889119906=

1198892V

1198891199062= and = 0

lowast = 0 = 119892(119906 V119889V

119889119906)

4 119906 = 119905 V = 119908 = 119890minus119910

119889V

119889119906=

119890minus119910

= 119891(119906 V) = 119892(119906 V)

119860119886

36

|119886| le 1119886 = 0 1

1 119906 = 119910 V = 119886minus1

119908 =

119889V

119889119906= 119886minus2

+ (119886 minus 1)119886minus3

= 119891(119906 V) 119886minus2 + (119886 minus 1)

119886minus3 = 119892(119906 V)

2 119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906=

119890minus119910

119886minus1

2+

+(119886 minus 1)119890


2+

= 119891(119906 V)

119890minus119910

119886minus1

2+

+(119886 minus 1)119890


2+

= 119892(119906 V)

3 119906 = 119910 V = 119905 119908 = 119905(2119886minus1)(119886minus1)

119889V

119889119906= 1 +

119905

119905(2119886minus1)(119886minus1)

= 119891(119906 V) 1 +119905

= 119892(119906 V)

11986037

1 119906 = 119910 V =119890minus119886 arctan

radic1 + 2

119908 =119890minus119886 arctan

(1 + 2)32

119889V

119889119906=

minus(119886 + )119890minus119886 arctan

(1 + 2)32

+119890minus119886 arctan

radic1 + 2

119890minus119886 arctan

(1 + 2)32

= 119891(119906 V) minus(119886 + )119890minus119886 arctan

(1 + 2)32


radic1 + 2

= 119892(119906 V)

2 119906 = 119910 + arctan V =119890minus119886 arctan

radic1 + 2

119908 =119890minus119886 arctan

(1 + 2)32

119889V

119889119906=


( + (1 + 2))radic1 +

2

+radic1 +


+ (1 + 2)


(1 + 2)32

= 119891(119906 V)minus(119886 + )119890

minus119886 arctan

( + (1 + 2))radic1 +

2

+radic1 +


+ (1 + 2)

= 119892(119906 V)

3 119906 = 119910 V = (1 + 1199052) 119908 =

119890119886 arctan 119905

32

119889V

119889119906= 2119905 +

(1 + 1199052)

119890119886 arctan 119905

32

= 119891(119906 V) 2119905 +(1 + 119905

2)

= 119892(119906 V)

11986038

2 119906 = 2119910 minus V = 2minus 4119909 119908 = 119909 minus 2119909

119889V

119889119906= minus2 minus

4119909

2 minus

119909 minus 2119909 = 119891(119906 V) minus2 minus4119909

2 minus = 119892(119906 V)

3 119906 = 119910 V =

(119905 minus 119909)22 119908 =

2

(119905 minus 119909)+

2

119889V

119889119906=

(119905 minus 119909)23minus

2(1 minus ) + 2(119905 minus 119909)

(119905 minus 119909)34

2

(119905 minus 119909)+

2

= 119891(119906 V)

(119905 minus 119909)23minus

2(1 minus ) + 2(119905 minus 119909)

(119905 minus 119909)34

= 119892(119906 V)

4 119906 = 119910 V =119905 minus 119909

119908 =

3

119889V

119889119906=

119905

2minus

(119905 minus 119909)

3

3

= 119891(119906 V) 119905

2minus

(119905 minus 119909)

3

= 119892(119906 V)

5 119906 = 119910 V =119909

radic1 + 2

119908 =1 + 2

11990933

+

11990923

119889V

119889119906=

(1 + 2minus 119909)

(1 + 2)32

+119909

radic1 + 2

1 + 2

11990933

+

11990923

= 119891(119906 V) (1 + 2minus 119909)

(1 + 2)32

+119909

radic1 + 2

= 119892(119906 V)


Table 2 Continued


11986039

1 119906 = 119910 V =1 + 2+ (119905 minus 119909)

2

(1 + 1199052 + 1199092)22

119908 =

(1 + 1199052 + 1199092)32

3

119889V

119889119906= minus

2(1 + 2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2

(1 + 1199052 + 1199092)32

3

= 119891 (119906 V)

minus2(1 +

2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2= 119892(119906 V)

2 119906 = minus119910 + arctan( sec119909) V = sec119909 + tan119909

radic1 + 2sec2119909

119908 = sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

119889V

119889119906=

(cos2119909 + 2) sec3119909

(( + 2(sin119909 minus ) sec119909) sec119909 minus )radic1 +

2sec2119909

+((1 +

2+ sin119909) cos119909 minus ( + sin119909)) sec3119909


2sec2119909 sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

= 119891(119906 V)

(cos2119909 + 2) sec3119909

(( + 2(sin119909 minus )sec119909)sec119909 minus )radic1 +

2sec2119909

+((1 +



2sec2119909= 119892(119906 V)

admitting the three-dimensional Lie algebra A119886236

|119886| le

1 119886 = 0 1 We arrive at the following set of invariants

119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906=

119890minus119910

119886minus1

2+

+(119886 minus 1) 119890


2+

(6)

This set involves three invariants 119906 V and 119908 which play therole of basis of invariants for the considered systemof second-order ODEs admitting the three-dimensional Lie algebra1198601198862

36 |119886| le 1 119886 = 0 1 This set is a complete set of invariants

for the system In the complete set one of the invariants isobtained by differentiation We state a result on invariantdifferentiation of invariants corresponding to a Lie algebraadmitted by any systemofODEswhich is alreadywell-knownfor scalar ODEs in the literature

Proposition 1 If 119906 and V are invariants of a Lie algebraadmitted by any system of ODEs then 119889V119889119906 is also itsinvariant

The proof is identical (except here we have many depen-dent variables) to the scalar case and uses the operatoridentity 119883119863

119905minus 119863119905119883 = minus(119863

119905120585)119863119905(see eg [17])

One can write119889V

119889119906=

119863119905V

119863119905119906

= 119863V (7)

where the invariant differentiation operator is

119863 = (119863119905119906)minus1

119863119905 (8)

There arises a natural question Can we construct a 119896th-order system from a basis of invariants corresponding to thesymmetry vectors of a second-order system We answer thisquestion

Suppose that we have a complete set of invariants repre-senting a second-order system of ODEs admitting a three-dimensional Lie algebra say for the above case 119860

1198862

36 |119886| le

1 119886 = 0 1 Now if we differentiate both of the highest-order invariants from this set with respect to another basicinvariant we get two new invariants and the total numberof invariants become six for this case including the previousfour These two new invariants play the role of third-orderinvariants representing the third-order system of ODEsadmitting the three-dimensional Lie algebra 119860

1198862

36 |119886| le

1 119886 = 0 1 We therefore have for this case

119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906= 119863V

119889119908

119889119906= 119863119908

1198892V

1198891199062= 1198632V

(9)


where119863 = 119890minus119910

(2+2)minus1

119863119905and119863119908 and119863

2V are third-orderinvariants We can thus construct the following third-ordersystem

119890minus119910

119909 minus 119890minus119910

( + )

22(2+ )

= 119891 (119906 V 119908 119863V)

(119890minus2119910

119886minus3

(minus5+ (2119886 minus 5)

3 minus (119886 minus 1)

4

+2 (119886 minus 1) 2)) ((

2+ )3

)minus1

+ (119890minus2119910

119886minus3

((119886 minus 1) ((119886 minus 5) 22+ (119886 minus 2)

3)

+2 119909 (2+)+

2((119886minus1) minus)

119910))

times ((2+ )3

)minus1

= 119892 (119906 V 119908 119863V)

(10)

We have constructed a system of two third-order ODEsadmitting the three-dimensional Lie algebra 119860

1198862

36 |119886| le

1 119886 = 0 1 from their complete set of invariants This sys-tem of ODEs is represented by a complete set of invari-ants which contains six functionally independent invariantsnamely 119906 V 119908 119863V 119863119908 and 119863

2V where three of them119906 V and 119908 are a basis of invariants and two of them119863119908 119863

2V are third-order invariantsNow if we repeat this process on the third-order invari-

ants we obtain eight invariants including the previous oneswhich form a complete set of invariants for the fourthprolonged group From these we can construct a system oftwo fourth-order ODEs admitting the three-dimensional Liealgebra 119860

1198862

36 |119886| le 1 119886 = 0 1 We can generalize this to

a system of two 119896th-order ODEs However there are somenonstandard cases to look at first These behave differentlyThere are four cases 119860

3

31 1198603

35 1198604

31 and 119860

1

38in which two

of them 1198603

31and 119860

3

35yield singular invariants and the other

two 1198604

31and 119860

1

38give a single equation and do not form a

system

21 Singular Case In both of the cases119860331

and1198603

35 we have

an extra condition due to which the rank of the coefficientmatrix on the solution manifold is less than the rank ofthe coefficient matrix on the generic manifold So singularinvariants are obtained in both of these cases

Consider 119860331

≃ 120597119909 119905120597119909 119910120597119909 We find the following set of

singular invariants

119906 = 119905 V = 119910 119908 = 119863V = = 0 (11)

where 119863 = 119863119905and the corresponding system are

= 119891 (119906 V 119863V) = 0 (12)

In both of the cases the singularity of the invariants isremoved in the third- and higher-order systems In the caseof 119860331 we get the following complete set of invariants

119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909 minus

119910 119863

3V =

119910

(13)

and the corresponding third-order system is

119909 =

119910

+

1

119891 (119906 V 119863V 119863

2V)

119910 = 119892 (119906 V 119863V 119863

2V)

(14)

In this case 119906 V and 119908 form a basis of invariants and herethe third-order invariants are 119908 and 119863

3V where 119863 = 119863119905 For

the other case 1198603

35 it is easy to see that

119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909

119863

3V =

119910

(15)

where again119863 = 119863119905 For each of the remaining cases1198604

31and

1198601

38 we obtain the same set of second-order invariants

119906 = 119909 V = 119910 119863V =

1198632V =

2minus

3

(16)

for which the highest-order invariant is only one and 119863 =

minus1

119863119905 So a system of second-order ODEs cannot be con-

structed Only a single scalar equation is obtained namely

=

+ 2119892 (119906 V 119863V) (17)

However in both of these cases this behavior changes forhigher orders In fact for 119860

4

31 we have the third-order

invariants

119908 =

11990912060110158401015840(119909)

4

minus3212060110158401015840(119909) +

2120601101584010158401015840

(119909)

5

1198633V =

119910

3+

(32minus

119909) minus 3

5

(18)

and for 119860138

the third-order invariants are

119908 =

119909

3minus

32

24 119863

3V =

119910

3+

(32minus

119909) minus 3

5

(19)

They give rise to a regular system of third-order ODE witha complete set of invariants having the same format as thesingular cases similar to (13)

We find that for a system of two second-order ODEsadmitting three-dimensional Lie algebras there are 29 classes


which are listed in Table 2 together with their second-orderdifferential invariants

In the case of a system of two third- and higher-orderODEs admitting three-dimensional Lie algebras there are31 canonical forms and these can be obtained by invariantdifferentiation as described above Note here that 119860

4

31and

1198601

38are admitted by a system of two third- and higher-order

ODEswhereas they donot forma systemof two second-orderODEs

We state these results in the form of a theorem

Theorem 2 A regular system of two 119896th-order ODEs (119896 ge 3)

admitting three-dimensional Lie algebras can be representedby a complete set of invariants which contains 2119896 completefunctionally independent invariants in one of the followingforms

(a) 119906 V 119908119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus2

119908

119889119906119896minus2119889119896minus1V

119889119906119896minus1

(b) 119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus3

119908

119889119906119896minus3

119889119896V

119889119906119896

(20)

In both (a) and (b) 119906 V and 119908 form a basis of invariantswhere the highest order invariants in (a) are 119889

119896minus2119908119889119906119896minus2

119889119896minus1V119889119906119896minus1 and in (b) they are 119889

119896minus3119908119889119906119896minus3

119889119896V119889119906119896

The total number of systems of two 119896th-order ODEshaving the above structure is thirty-one in which form (b)corresponds to the cases 1198603

31 1198603

35 1198604

31 and 119860

1

38and (a) to

the remainder

3 Integrability

There are basically two approaches to integrability using Liepoint symmetries One is that of successive reduction of orderusing differential invariants for scalar ODEs The other isthat of canonical forms which may apply to scalar as wellas system of ODEs although the latter is not straightfor-ward However the canonical forms approach only applieswhen equations are classified according to the symmetryLie algebras In Table 2 we have given the complete set ofsecond-order differential invariants of three-dimensional Liealgebras admitted by systems of two second-order ODEs andalso classified systems of two second-order ODEs for theirpoint symmetries together with their canonical forms Weprovide two approaches for the integration procedure on thebasis of these canonical formsThe first is a general approachdepending on the subdivision of the canonical forms based ongeneral observationsThe second is the differential invariantsapproach based on the representation of the canonical formsin terms of first-order differential and algebraic invariants

31 General Integration Approach In this approach we judi-ciously subdivide all the canonical forms into four typesTheyare as followsType I

1198603

31⋍ 120597119909 119905120597119909 119910120597119909 = 119891 (119905 119910 ) = 0 (21)

Type II

1198603

35⋍ 120597119909 119905120597119909 119909120597119909 = 119892 (119905 119910 ) = 0 (22)

Type III

1198604

32⋍ 120597119909 119905120597119905+ 119909120597119909 119905120597119909

119905 = 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198603

34⋍ 120597119909 119905120597119909 minus120597119905+ 119909120597119909

119890119905 = 119891 (119910 )

= 119892 (119910 )

1198604

35⋍ 120597119909 119905120597119909 119909120597119909+ 120597119910

119890minus119910

= 119891 (119905 ) = 119892 (119905 )

1198601198863

36⋍ 120597119909 119905120597119909 (1 minus 119886) 119905120597

119905+ 119909120597119909

119905(2119886minus1)(119886minus1)

= 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198601198863

37⋍ 120597119909 119905120597119909 minus (1 + 119905

2) 120597119905+ (119886 minus 119905) 119909120597

119909

minus32

119890119886 tan 119905

= 119891 (119910 (1 + 1199052))

2119905 +(1 + 119905

2)

= 119892 (119910 (1 + 119905

2))

(23)

Type IV Remaining canonical forms except those givenabove

We notice that the Types I and II systems comprise linearequations The Type I system is completely integrable Alsothat the Type II system is uncoupled and depends on twoquadratures of the 119910 equation for its solution Further weobserve that the Type III has equations that comprise scalarsecond-order ODEs in the dependent variable 119910 In view ofthese observations we are led to the following definitions

Definition 3 Asystemof two second-orderODEs is said to bepartially uncoupled if one of its equations constitutes a scalarsecond-order equation in one of the dependent variables andthe other has both

It is easily seen that the Types I to III equations arepartially uncoupled

Definition 4 A partially uncoupled system of two second-order ODEs is partially linear if one of its equations formsa scalar linear second-order equation in its own right

We see that the Types I and II systems are partially linear

Definition 5 A system of two second-order ODEs is said tobe partially linearizable by point transformation if it can betransformed via an invertible transformation to a partiallylinear form


The following system

= 4 + 3119910 + 119910

3= 0 (24)

is partially linearizable by a point transformation as theequation in 119910 is the well-known Painleve-Ince equationwhich is linearizable via a point transformation

Let 119883119894 be a basis of a three-dimensional algebra and

let the rank of the coefficient matrix associated with theseoperators be 119903

We formulate the following theorems the proofs of whichare evident

Theorem 6 A system of two second-order ODEs is partiallylinearizable by point transformation if and only if it admits theLie algebra 119860

31or 11986035

with rank 119903 = 1 in both cases

Theorem 7 A system of two second-order ODEs can bereduced to a partially uncoupled system by point transfor-mation if and only if it admits one of the Lie algebras1198604

32 1198603

34 1198604

35 119860119886336 and 119860

1198863

37with rank 119903 = 2 in all cases

The proofs follow from the above listed Types I to IIIalgebras and representative equations

Generally we can integrate Type I completely TypesII and III can be integrated when the associated second-order equation in the system is integrable Moreover onlyparticular cases of Type IV can be integrated depending onthe given Lie algebra and corresponding system of ODEsSome of these difficulties were also pointed out for systemsof two second-order ODEs that admit four symmetries[12] Here we further discuss all the cases in detail beforeembarking on a further systematic study that captures themain results

Type I is trivially integrable by quadraturesThe Type II system is integrable if the associated second-

order ODE in 119910 admits a two-dimensional Lie algebra andLiersquos four canonical forms for scalar second-order ODEsbecome applicable So we have Lie reducibility here

In the case of Type III one requires one more symmetryfor the associated equation in 119910 as it already has translationsin 119905 symmetry As an example we consider the system

= 119890minus11990511991034 =

2119910minus1

minus 1199102 (25)

The system (25) has symmetry generators 1198831

= 120597119909 1198832

=

119905120597119909 1198833= minus120597119905+ 119909120597119909and its Lie algebra is three-dimensional

The associated equation in 119910 is = 2119910minus1

minus1199102 Its Lie algebra

is two-dimensional spanned by 1198841= 120597119905 1198842= 119905120597119905minus 2119910120597

119910 By

using the transformation

119905 = 119910 119909 = 119909 119910 = 119905 (26)

the associated equation in transformed form becomes =

1199052

3

minus 119905minus1Wehave119884

1= 120597119910 1198842= 119910120597119910minus 2119905120597119905 which is Type

III in Liersquos classification of scalar second-order ODEs Solvingit in transformed form by Liersquos method and reverting backto the original variables we obtain 119910 = 2119888

1

2sech21198881(119905 minus 119888

2)

where 1198881and 1198882are constants By using the value of119910 in system

(25) and then solving for 119909 we deduce

119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

times tanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(27)

In general the Type III systems of second-order ODEs areintegrable if the associated second-order ODEs in 119910 haveanother symmetry generator besides translations in 119905 so thatLiersquos classification of scalar second-order ODEs admittingtwo-dimensional algebras is applicable here as well

Theorem 8 If a singular system of two second-order ODEsadmits the solvable symmetry algebra 119860

3

31 then it is partially

linearizable via point transformation and its general solutioncan be trivially obtained by quadratures

The proof of this follows from the Type I canonical form

Theorem 9 If a singular system of two second-order ODEsadmits the three-dimensional solvable symmetry algebra 119860

3

35

then it can be reduced to a partially uncoupled system andits general solution can be obtained by quadratures from thegeneral solution of the associated transformed scalar second-order ODE

The proof here follows from the Type II canonical formFor systems that fall in the Type IV category one requires

more insight and the examples given in [12] for systems thatadmit four symmetries become relevant Their integrabilityneeds more than just the reductions via invariants Thiscategory also has the nonsolvable algebras We consider thisin detail next

32 Differential Invariant Approach In this approach we usethe basis of invariants representing all the canonical formswhich we have given in Table 2 For each regular canonicalform we find the reduced system in terms of a first-orderequation and an algebraic equation based on the basis ofinvariants These can easily be seen from Table 2

We have subdivided all the reduced systems in two maincategories namely Category A for those admitting three-dimensional solvable Lie algebras and Category B for thoseadmitting three-dimensional nonsolvable Lie algebras TheTypes I to III of the previous subsection all fall in CategoryA The systems admitting nonsolvable algebras compriseCategory B with the rest firmly falling in Category A It isinteresting that in both categories uncoupled cases arisewhen the choice of taking one of the second-order invariant119908 is unique

We present the integration strategy for Category A forregular systems The systems of Category B are not amenableto such integration strategies This is quite straightforwardto work out if one commences with the simplest system inthis category We further subdivide Category A into coupledand uncoupled systems All partially linear cases fall in the


uncoupled systems Here we give details of integration foreach case of Category A which depends on the assumptionthat we can solve 119889V119889119906 = ℎ in both the coupled and theuncoupled systems At the end of this section we presentsome physical examples with detailed calculations

Note here that in both the uncoupled and coupled casesfor ease and simplicity of the calculations we only mentionthe relevant algebra of the considered system and variables119906 V and 119908 as well as 119889V119889119906 in precise form The detailsare given in Table 2 In the following we assume that V =

120582(119906 1198881) is the explicit solution of each of the first order ODEs

119889V119889119906 = ℎ(119906 V) listed in Table 2 The argument presentedbelow is similar to that used in the discussion of reductionto quadrature for the scalar ODEs (see [1ndash3]) Moreover119891 119892 ℎ 119896 120601 120595 and 120582 are taken as arbitrary functionsand 119888

1 1198882 and 119888

3as well as 119888

4are arbitrary integration

constants

321 Uncoupled Cases

1198604

32 Let V = 120582(119906 119888

1) Then by using the values of 119906 and V in

Table 2 we have

119905 = 120582 (119910 1198881) (28)

Thus we can obtain a further quadrature in 119910 Therefore theequation 119905 = 119891(119910 119905) is solvable by two quadratures uponinsertion of the 119910 solution

1198601

33 Let V = 120582(119906 119888

1) Then by invoking the values of 119906 and V

as listed in Table 2 we obtain

= 120582 (119910 1198881) (29)

After some manipulations with 119908 = 119891(119906 V) we find

=minus1

int 120582 (119910 1198881) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2

(30)

By using (30) in (29) we determine

120582 (119910 1198881) int120582 (119910 119888

1) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2 = minus1 (31)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (32)

Utilizing (32) in (30) after some further calculations wededuce

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (33)

Hence (32) and (33) are the solutions of the system of ODEsadmitting 119860

1

33

1198602

33 Let V = 120582(119906 119888

1) By using the values of 119906 and V as listed

in Table 2 we arrive at

= plusmnradic2 + 22120582 (119910 119888

1) (34)

After calculations with 119908 = 119891(119906 V) we find

= minus int119891 (119910 120582 (119910 1198881)) 119889119910 + 119888

2 (35)

Invoking (35) in (34) and solving for we obtain

=2

(int119891(119910 120582(119910 1198881))119889119910 + 119888

2)2

minus 2120582 (119910 1198881)

(36)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (37)

After some computations and using (37) in (35) we deter-mine

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (38)

It is hence the case that (37) and (38) are the solutions of thesystem of ODEs corresponding to 119860

2

33

1198603

34 Let V = 120582(119906 119888

1) Using the values of 119906 and V we obtain

= 120582 (119910 1198881) (39)

which implies that

119910 = 120601 (119905 1198881 1198882) (40)

Now utilizing (39) and (40) in 119908 = 119891(119906 V) we deduce that

119909 = intint 119890minus119905119891 (120601 120601

119905) 119889119905 119889119905 + 119888

3119905 + 1198884 (41)

which results in

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (42)

Now (40) and (42) are the solutions of the system of ODEspossessing 119860

3

34

1198601

35 Let V = 120582(119906 119888

1) Utilizing the values of 119906 and V as listed

we obtain

= 120582 (119910 1198881) (43)


int119889119910

1198882exp int119891 (119910 120582 (119906 119888

1)) 119889119910

= 119905 + 1198883 (44)

Inverting we have

119910 = 120601 (119905 1198881 1198882 1198883) (45)

By using (45) in (43) we have that

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (46)

Thus (45) and (46) are the solutions of the system of ODEscorresponding to 119860

1

35

For119860435

1198601198863

36 and119860

1198863

37 the integration procedure works

the same as in the case 1198603

34


322 Coupled Cases

1198601

31 Let V = 120582(119906 119888

1) Using the values of 119906 and V as listed in

Table 2 we obtain

= 120582 ( 1198881) (47)

After calculations with 119908 = 119891(119906 V) we arrive at

119909 = 120601 (119905 1198881 1198882 1198883) (48)

By utilizing (48) in (47) in the same manner as in theprevious cases we find that

119910 = 120595 (119905 1198881 1198882 1198883 1198884) (49)

Hence (48) and (49) are the solutions of the system of ODEscorresponding to 119860

1

31

1198602

31 The procedure of integration is the same as in the case

1198601

31

1198601

32 Let V = 120582(119906 119888


Table 2 we determine

= 120582 (119890119910 1198881) (50)


= exp int ((120582 (119890119910 1198881) + 119890

119910119892 (119890119910 120582 (119890

119910 1198881)))

times119891 (119890119910 120582 (119890

119910 1198881)) 119889 (119890

119910))

times (119890119910(1+120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881))))minus1

(51)

From the second equation of the system in Table 2 and somemanipulations with (50) we find that

119889 (119890119910)

119889119905

=2119890119910120582 (119890119910 1198881) (1 + 120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881)))

(120582 (119890119910 1198881) + 119890119910119892 (119890119910 120582 (119890119910 119888

1)))

(52)

Some further computations give

119890119910= 120601 (119910 + 119888

2) (53)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (54)

Using (53) and (54) in (51) we find

119909 = 119896 (119905 1198881 1198882 1198883 1198884) (55)

Therefore (54) and (55) are the solutions of the system ofODEs corresponding to 119860

1

32

1198602

32The procedure of integration is the same as the case1198601

35

1198603

32 The procedure of integration is similar to the case 119860

1

33

1198603

33 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 (119910 minus

1

1198881) (56)

Using the chain rule and some manipulations with 119908 =

119891(119906 V) we arrive at

119910 = 120601(119910 minus1

1198881 1198882) (57)

Further simplifications give

=1

119910 minus 120601minus1 (119910 1198881 1198882) (58)

Utilizing (57) and (58) in (56) we find that

(119910 minus 120601minus1

(119910 1198881 1198882)) 120582 (120601

minus1(119910 1198881 1198882) 1198881) = 1 (59)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (60)

Invoking (60) in (58) we have

119909 = 119896 (119905 1198881 1198882 1198883 1198884) (61)

Thus (60) and (61) are the solutions of the system of ODEspossessing 119860

3

33

1198601


1

33

1198602


33

1198602

35 Let V = 120582(119906 119888


in Table 2 we obtain

119890119910= 120582 ( 119888

1) (62)

By the chain rule and some simplifications with 119908 = 119891(119906 V)we find

= 120601 (119910 1198881 1198882) (63)

Substituting (63) in (62) and further manipulations give

119910 = 120595 (119909 1198881 1198882 1198883) (64)

Invoking this in (63) we deduce that

119909 = ℎ (119905 1198881 1198882 1198883 1198884) (65)

Now after substitution of (65) (64) takes the form

119910 = 120595 (119905 1198881 1198882 1198883 1198884) (66)

Here (65) and (66) are the solutions of the system of ODEscorresponding to 119860

2

35


1198601198861


33

1198601198862

36 The procedure of integration is again similar to the case

1198601

32

1198601198861

37 The procedure of integration corresponds to the case

1198601

33and some details are given in the case 119860

1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)

After manipulations with 119908 = 119891(119906 V) we find

= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)

Invoking (68) in (67) we determine

= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)

Using (69) and (72) in (70) we deduce

119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)

By invoking (74) in (68) we have that

119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37

In the view of the preceding reductions we have thefollowing theorem

Theorem 10 If a regular system of two second-order ODEsadmits a three-dimensional solvable symmetry algebra thenthe general solution of the regular system can be obtainedby quadratures from the general solution of the invariantrepresentation of the regular system given by the first-orderordinary differential equation and algebraic equation

Theorem 11 If a system of two 119896th-order (119896 ge 3)ODEs possessa three-dimensional solvable symmetry algebra then thegeneral solution of the system can be deduced by quadraturesfrom the general solution of the invariant representation givenby one of the following systems

(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)

The proof follows from Theorem 2 and the discussionprecedingTheorem 10

323 Illustrative Examples Here we present some familiarphysical examples to illustrate the above general integrationprocedure which we have given in detail for each of the caseof system of ODEs admitting solvable three-dimensional Liealgebras We consider two different cases namely one fromthe uncoupled cases and one from the coupled cases We alsopresent a case of nonsolvable Lie algebras Here we observeinteresting results rather than integration

(i) Consider the system (25) once again from a new angleWe have

= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603

34in our classification From Table 2 the first-

order differential invariants for this case are

119906 = 119910 V = (78)

and the reduced system of first-order equation and algebraicequation is

119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)

The solution of 119889V119889119906 in system (79) after substituting thevalue of 119892 from (25) gives

V = plusmn119906radic2 (119888lowast1

minus 119906) (80)


where 119888lowast

1is constant Reverting back to the original variables

and after solving for 119910 we obtain

119910 = 21198881

2sech21198881(119905 minus 1198882) (81)

where 1198881= radic119888lowast12 and 119888

2are constants Substituting the values

in 119908 and solving for 119909 we find

119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)

Both (81) and (82) provide the solution of system (25)(ii) The classical Kepler problem in reduced Cartesian

coordinates is

119903 + 120583119903

1199033= 0 (83)

where 120583 is a constant In polar coordinates they are

119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)

It has the three-dimensional Lie algebra of symmetries [18]spanned by

1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)

By using the transformation

119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)

the system (84) transforms into

+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)

with associated transformed Lie algebra of symmetries

1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)

By utilizing the values of 119891 and 119892 from (87) we have that

119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)

Now the solution of 119889V119889119906 in the system (91) is

1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)

In 119909 and 119910 variables after some simplification (92) and 119908

from (91) give

119889 (119890(23)119910

)

119889119905= plusmnradic

4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)

where ℎ and 119896 are constants Reverting back to the originalvariables we deduce

119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)

The term on the left-hand side of (95) is the angularmomentum and here it is constant from the calculations anddenoted by 119871119898 Performing a simple integration procedurewe arrive at the equation of the orbit namely

119889120579 =119889119904

∓radic(ℎlowast + 211989821205831199041198712 minus 1199042)

(96)

where 119904(120579) = 1119903(120579) and ℎlowast

= 4ℎ91198964 This corresponds to

that presented in [19](iii) Consider a case of nonsolvable Lie algebras namely

the generalized Ermakov system in polar coordinates of theform

119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)

which was considered in [20] This system has the threesymmetries

1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)

and forms the Lie algebra sl(2119877) which is 1198604

38in our

classification The first-order differential invariants for thiscase are 119906 = 119910 V = (119905 minus 119909) and the reduced systemof first-order equation and an algebraic equation is 119889V119889119906 =

119892(119906 V) 119908 = 119891(119906 V) By using these and considering theangular momentum as constant 119871 = 119898119903

2 we find that

119889V119889119906 = minusVℎ(119906 119871119898) where 119892(119906 V) = minusVℎ(119906 119871119898) In theoriginal variables it is

119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)

where ℎ(120579) = ℎ(119906 119871119898) Here 119872 is the Ermakov invariantThe classical integrability was discussed in [20]


4 Conclusion

The integrability of systems of ordinary differential equations(ODEs) which admit symmetry algebras has been of interestin the literature The traditional symmetry reductions do notwork for systems of ODEs as in the case of scalar ODEs (seeeg [5 12 17]) Here we have focused on systems of two119896th-order (119896 ge 2) ODEs that admit three-dimensional Liealgebras

By utilizing the realizations given recently we haveprovided a systematic approach for finding a complete setof differential invariants including basis of invariants corre-sponding to vector fields in three variables for systems of two119896th-order (119896 ge 2) ODEs which admit three-dimensional Liealgebras In particular we presented a complete list of second-order differential invariants for systems of two second-orderODEs that admit three-dimensional Lie algebras We haveshown how invariant differentiations provide higher-orderdifferential invariants We have given a complete list ofinvariant characterizations of canonical forms for systemsof two second-order ODEs that admit real Lie algebras ofdimension three These are new in the literature In additionwe have given a procedure for the construction of two 119896th-order ODEs from their associated complete set of invariantsWe have shown that there are 29 classes for the case of119896 = 2 and 31 classes for the case of 119896 ge 3 of such ODEsFurthermore we have given a discussion of those cases inwhich regular systems of two second-order ODEs are notobtained for three-dimensional Lie algebras We also havegiven a brief discussion on the singularity for canonical formsfor a system of two second-order ODEs admitting-threedimensional Lie algebras and found two cases of singularinvariants namely those involving the algebras1198603

31and119860

3

35

We have provided an integration procedure for thecanonical forms for a system of two second-order ODEswhich possess three-dimensional Lie algebras This proce-dure is composed of two approaches namely one whichdepends on general observations of canonical forms theother depends on basis of differential invariants and thesolution procedure is applicable only for those cases whichadmit three-dimensional solvable Lie algebrasThe latter wasextended in a natural way to 119896th-order (119896 ge 3) systemsthat admit three-dimensional Lie algebras Specifically in thesecond approach we obtain the reducibility for canonicalforms for systems of two second-order ODEs which havethree-dimensional Lie algebras in terms of basis of invariantsthese are given in Table 2Then we gave details of integrationfor each case of solvable Lie algebras in a general manner

We also have obtained two cases of partial linearizationnamely Types I and II It was noted that the Type I systemis trivially integrable We have further shown how theassociated scalar second-order ODEs in the Types II to IIIcan be solved in terms of quadratures resorting to the classicalLie table for integrability of scalar ODEs This was illustratedby means of an example There are certain Type IV systemswhich have complications in their integrability in that theyadmit nonsolvable algebras

We have presented familiar physical systems taken fromthe literature namely the classical Kepler problem and

the generalized Ermakov systems that give rise to closedtrajectories

It would be of interest to further pursue integrabilityproperties of Type IV systems having nonsolvable Lie alge-bras Also worthy of investigation are systems that admithigher number of symmetries such as those studied in [12 16]We return to these in the future

Remarks

(i) In the tables 119860 is used as a place holder for therelevant real Lie algebras 119860

119886119887119899

119894119895 (the 119895th algebra of

dimension 119894 with superscripts 119886 119887 if any indicatesparameters on which the Lie algebra depends alsothe column 119873 in the tables gives the algebra realiza-tions the realization is referred to by a superscript119899 eg 119860

1

31) Further we use the notation 120597

119905=

120597120597119905 120597119909

= 120597120597119909 120597119910

= 120597120597119910 Finally the elements ofa basis of a given Lie algebra are named119883

119894 where 119894 is

less than or equal to the dimension of the underlyingreal Lie algebra Here 119891 119892 ℎ 119896 and 120601 are arbitraryfunctions

(ii) For the analysis of the invariant representations itis observed that for each of the cases 119860

4

31and

1198601

38 with the same set of invariants a single equa-

tion is determined We find the differential invari-ants in both cases However the canonical formis not obtained Consider for example 119860

4

31≃

120597119905 119909120597119905 120601(119909)120597

119905 12060110158401015840(119909) = 0 For this we find the fol-

lowing invariants 119906 = 119909 V = 119910 119889V119889119906 =

1198892V1198891199062 = minus

3+2 and the corresponding sin-

gle equation namelyminus3+2= 119892(119906 V 119889V119889119906)

For each of the cases 1198603

31and 119860

3

35 due to the extra

condition the rank of the coefficient matrix on thesolution manifold is less than the rank of genericcoefficient matrix So only a set of singular invariantsare obtained for these cases Consider for example1198603

31≃ 120597119909 119905120597119909 119910120597119909 For this we find the following

set of singular invariants 119906 = 119905 V = 119910 119889V119889119906 =

119908 = and 119889V119889119906 = = 0 (see Table 2) Thetwo cases 1198604

31and 119860

1

38are not given in the invariant

representations listed in Table 2(iii) The algebras 119860

3

31and 119860

3

35which are referred to by

lowast give rise to singular invariants when associatedwith systems of two second-order ODEs as given inTable 2 We do however arrive at regular invariantsassociated with systems of two third-order ODEs forthese cases

(iv) The algebras 1198604

31and 119860

1

38do not correspond to

systems of two second-order ODEs and so are notpresented in Table 2

Acknowledgments

M Ayub and M Khan are grateful to the Higher EducationCommission (HEC) of Pakistan for financial support (via


Indigenous Scholarship and IRSIP program) Also F MMahomed thanks the HEC for granting him a visiting pro-fessorship and the Department of Mathematics at the Quaid-i-Azam University Islamabad for its hospitality during thetime this work was initiated M Ayub also thanks theCentre for Differential Equations Continuum Mechanicsand Applications School of Computational and AppliedMathematics University of the Witwatersrand South Africafor its hospitality during the time this project was completedFinally F MMahomed is grateful to the NRF of South Africafor research funding Parts of this work was presented at theSDEA2012 Jan Conference

References

[1] S Lie ldquoKlassiffikation und Integration von gewonlichen Differ-entialgleichungen zwischen x y die eine Gruppe von Transfor-mationen gestatenrdquo Archiv der Mathematik vol 8 no 9 p 1871883

[2] S Lie Vorlesungen Uber Differentialgleichungen Mit BekanntenInfinitesimalen Transformationen B G Teubner Leipzig Ger-many 1891

[3] FMMahomed and P G L Leach ldquoLie algebras associated withscalar second-order ordinary differential equationsrdquo Journal ofMathematical Physics vol 30 no 12 pp 2770ndash2777 1989

[4] F M Mahomed and P G L Leach ldquoSymmetry Lie algebras of119899th order ordinary differential equationsrdquo Journal ofMathemat-ical Analysis and Applications vol 151 no 1 pp 80ndash107 1990

[5] C Wafo Soh and F M Mahomed ldquoReduction of order forsystems of ordinary differential equationsrdquo Journal of NonlinearMathematical Physics vol 11 no 1 pp 13ndash20 2004

[6] R O Popovych V M Boyko M O Nesterenko and M WLutfullin ldquoRealizations of real low-dimensional Lie algebrasrdquoJournal of Physics vol 36 no 26 pp 7337ndash7360 2003

[7] S Lie Theories der Transformation Gruppen Chelsea Pub-lishing New York NY USA 2nd edition 1970 Originallypublished in Leipzig 1888 1890 1893

[8] S Lie ldquoTheorie der transformationsgruppenrdquo MathematischeAnnalen vol 16 no 4 pp 441ndash528 1880

[9] S Lie ldquoTheorie der transformationsgruppenrdquo in GesammetleAbhandlungen vol 6 pp 1ndash94 B G Teubner Leipzig Ger-many 1927

[10] L Bianchi Lezioni Sulla Teoria dei Gruppi Continui Finiti diTransformazioni Enirco Spoerri Pisa Italy 1918

[11] G M Mubarakzjanov ldquoOn solvable Lie algebrasrdquo IzvestijaVyssih Ucebnyh Zavedeniı Matematika vol 1 no 32 pp 114ndash123 1963 (Russian)

[12] CWafo Soh and FMMahomed ldquoCanonical forms for systemsof two second-order ordinary differential equationsrdquo Journal ofPhysics vol 34 no 13 pp 2883ndash2911 2001

[13] A V Aminova and N A M Aminov ldquoProjective geometry ofsystems of differential equations general conceptionsrdquo Tensorvol 62 no 1 pp 65ndash86 2000

[14] P G L Leach ldquoA further note on the Henon-Heiles problemrdquoJournal ofMathematical Physics vol 22 no 4 pp 679ndash682 1981

[15] P G L Leach and V M Gorringe ldquoThe relationship betweenthe symmetries of and the existence of conserved vectors forthe equation + 119891(119903)119871 + 119892(119903) = 0rdquo Journal of Physics vol 23no 13 pp 2765ndash2774 1990

[16] O Gaponova and M Nesterenko ldquoSystems of second-orderODEs invariant with respect to low-dimensional Lie algebrasrdquoPhysics AUC vol 16 part 2 pp 238ndash256 2006

[17] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[18] V Gorringe VM and P G L Leach ldquoKeplerrsquos third law and theoscillator isochronismrdquo American Journal of Physics vol 61 pp991ndash995 1993

[19] N H Ibragimov FMMahomed D PMason andD SherwellDi Erential Equations and Chaos New Age International NewDelhi India 1996

[20] K S Govinder and P G L Leach ldquoIntegrability of generalizedErmakov systemsrdquo Journal of Physics vol 27 no 12 pp 4153ndash4156 1994

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


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Stochastic AnalysisInternational Journal of


Systems of two second-order ODEs arise frequently inseveral applications in classical fluid and quantum mechan-ics as well as in general relativity Hence these have beenstudied vigorously over the years A sample of the literature asfar as symmetry Lie algebras of ODEs are concerned can befound in [13ndash15] In particular the classification of systemsof two second-order ODEs admitting real four-dimensionalLie algebras was attempted in [12] Gaponova andNesterenko[16] gave a classification of system of two second-order ODEsadmitting real three- and four-dimensional Lie algebras byusing the results of [6]

In the this paper by utilizing realizations given in [6]a systematic procedure for finding a complete set of 119896th-order (119896 ge 2) differential invariants including basis ofinvariants corresponding to vector fields in three variablesfor three-dimensional Lie algebras is presented In additionwe give a procedure for the construction of two 119896th-orderODEs possessing three-dimensional Lie algebras from theassociated complete set of invariants We show that thereare 29 classes for 119896 = 2 and 31 classes for 119896 ge 3Moreover we mention those cases in which regular systemsof two second-order ODEs are not obtained for three-dimensional Lie algebras We present a brief discussion onthe singular representation of canonical forms for systemsof two second-order ODEs admitting three-dimensional Liealgebras Furthermore we provide an integration procedurefor canonical forms for systems of two second-order ODEsadmitting three-dimensional Lie algebras This procedure iscomposed of two approaches namely one which depends ongeneral observations of canonical forms which admit three-dimensional Lie algebras the second depends on a completeset of differential invariants and is applicable only for thosecases which admit three-dimensional solvable Lie algebrasWe show by familiar physical examples how these integrationapproaches work In the second approach for solvable Liealgebras we analyze integrability in terms of integration ofthe invariant representations We prove that if a system oftwo second-order ODEs admits a solvable three-dimensionalLie algebra then its general solution can be deduced from apartially linear partially coupled or invariant reduced systemof equations A natural extension of this result is stated for asystem of two 119896th-order (119896 ge 3) ODEs

In Table 1 for ease of reference we list the realizations ofthree-dimensional Lie algebras as given in [6] In the secondsection for the Lie algebras of vector fields considered wediscuss and deduce a complete set of differential invariantsincluding basis of invariants with the construction of thecorresponding systems of two 119896th-order ODEs Moreoverfor systems of two second-order ODEs admitting three-dimensional Lie algebras we discuss the singular represen-tation and those cases in which we do not obtain systemsIn the end in Table 2 we give a complete list of second-order differential invariants corresponding to vector fields inthree variables of three-dimensional real Lie algebras and theassociated complete list of canonical forms for systems of twosecond-order ODEs In Section 3 we provide an integrationprocedure for the canonical forms of systems of two 119896th-order(119896 ge 2) ODEs admitting three-dimensional Lie algebrasThisis stated as theorems for systems admitting solvable algebras

Reduction of canonical forms for systems of two second-orderODEs into invariant systems of first-order and algebraicequations are given in Table 2 The paper ends with a briefconclusion

2 Invariants and Systems of ODEs

When one calculates the symmetries of a given differentialequation one finds the generators in the form of vectorfields and then computes the Lie brackets to get the structureconstants of the particular Lie algebra one has found We canstart from a given Lie algebra with a set of structure constantsand ask which vector fields in at most three variables satisfythe given Lie bracket relations with none of the vectorfields vanishing We thus ask for possible realizations ofthe given Lie algebra We make use of the realizationsof three-dimensional Lie algebras in the (1+2)-dimensionalspace given in [6] We utilize these realizations in order toobtain differential invariants and invariant representationsfor system of two second-order and higher-order ODEsExtensions to higher order invariants are given to understandthe behavior of such invariantsThis is the inverse problem insymmetry analysis and was initiated by Lie

For completeness and ease of reference we list therealizations of three-dimensional real algebras as given in [6]in Table 1 However we mostly keep the format of [12]

Differential invariants play an important role in thestructure and reduction of differential equations Differen-tial invariants can be obtained by various approaches andwe refer for example to [16] for realizations in (1+2)-dimensional space Differential invariants obtained by var-ious approaches corresponding to three-dimensional Liealgebras have very interesting properties In the investigationof the invariant approach for a system of two second-orderODEs admitting three-dimensional Lie algebras it is foundthat for a regular system ofODEs there are four independentinvariants three of which play the role of basis and the fourthone is derived from the remaining three

For the case of a regular system of two second-orderODEs admitting three-dimensional Lie algebras there arefour functionally independent invariants For example con-sider the system of two second-order ODEs

2

= 119891(119910

)

2minus

3

= 119892(119910

) (1)

admitting the three-dimensional Lie algebra 1198603

32 We have

the four independent invariants

1199081= 119910 119908

2=

119908

3=

2 119908

4=

2minus

3

(2)

Here the invariants1199083and 119908

4are of order two that is equal

to the order of the system of ODEs that is constructible




11986031

1 120597119905 120597119909 120597119910

3

2 120597119909 120597119910 119905120597119909+ 119891(119905)120597

119910 2

3 120597119909 119905120597119909 119910120597119909

1

4 120597119905 119909120597119905 120601(119909)120597

119905 12060110158401015840(119909) = 0 1

11986032

[1198831 1198832] = 119883

1

1 120597119905 119905120597119905+ 120597119910 120597119909

3

2 120597119905 119905120597119905+ 119910120597119909 120597119909

2

3 120597119905 119905120597119905 120597119909

2

4 120597119909 119905120597119905+ 119909120597119909 119905120597119909

2

11986033

[1198832 1198833] = 119883

1

1 120597119905 120597119909 119909120597119905

2

2 120597119905 120597119909 119909120597119905+ 119910120597119909

2

3 120597119905 120597119909 119909120597119905+ 120597119910

3

11986034

1 120597119905 120597119909 (119905 + 119909)120597

119905+ 119909120597119909

2

[1198831 1198833] = 119883

12 120597

119905 120597119909 (119905 + 119909)120597

119905+ 119909120597119909+ 120597119910

3

[1198832 1198833] = 119883

1+ 1198832

3 120597119909 119905120597119909 minus120597119905+ 119909120597119909

2

11986035

[1198831 1198833] = 119883

1

[1198832 1198833] = 119883

2

1 120597119905 120597119909 119905120597119905+ 119909120597119909

2

2 120597119905 120597119909 119905120597119905+ 119909120597119909+ 120597119910

3

3 120597119909 119905120597119909 119909120597119909

1

4 120597119909 119905120597119909 119909120597119909+ 120597119910

2

119860119886

36 |119886| le 1 119886 = 0 1 1 120597

119905 120597119909 119905120597119905+ 119886119909120597

1199092

[1198831 1198833] = 119883

12 120597

119905 120597119909 119905120597119905+ 119886119909120597

119909+ 120597119910

3

[1198832 1198833] = 119886119883

23 120597

119909 119905120597119909 (1 minus 119886)119905120597

119905+ 119909120597119909

2

119860119886

37 119886 ge 0 1 120597

119905 120597119909 (119886119905 + 119909)120597

119905+ (119886119909 minus 119905)120597

1199092

[1198831 1198833] = 119886119883

1minus 1198832

2 120597119905 120597119909 (119886119905 + 119909)120597

119905+ (119886119909 minus 119905)120597

119909+ 120597119910

3

[1198832 1198833] = 119883

1+ 1198861198832

3 120597119909 119905120597119909 minus(1 + 119905

2)120597119905+ (119886 minus 119905)119909120597

1199092

11986038

sl(2 119877)

[1198831 1198832] = 119883

1

[1198832 1198833] = 119883

3

[1198833 1198831] = minus2119883

2

1 120597119905 119905120597119905 1199052120597119905

1

2 120597119905 119905120597119905+ 119909120597119909 1199052120597119905+ 2119905119909120597

119909+ 119909120597119910

3

3 120597119905+ 120597119909 119905120597119905+ 119909120597119909 1199052120597119905+ 1199092120597119909

2

4 minus1199051205971199091

2(minus119905120597119905+ 119909120597119909) 119909120597119905

2

5 120597119905 119905120597119905+ 119909120597119909 (1199052minus 1199092)120597119905+ 2119905119909120597

1199092

11986039

so(3)[1198831 1198832] = 119883

3

1 (1 + 1199052)120597119905+ 119909119905120597

119909 119909120597119905minus 119905120597119909 minus119909119905120597

119905minus (1 + 119909

2)120597119909

2

[1198833 1198831] = 119883

2

[1198832 1198833] = 119883

1


119909+ sin 119905 sec119909120597

119910 minus cos 119905 tan119909120597

119905+ sin 119905120597

119909+ cos 119905 sec119909120597

119910 120597119905

3


2

minus22

3

= 119891(119910

2

2minus

3)

minus119910

4

+

1199092minus 3 + 3

2

5

= 119892(119910

2

2minus

3)

(3)


1199081= 119910 119908

2=

119908

3=

2 119908

4=

2minus

3

1199085=

119909

2

minus22

3 119908

6= minus

119910

4

+

1199092minus 3 + 3

2

5

(4)


5and 119908




= 119891 (119890

119910 119886minus1

)

119890minus119910

119886minus1

2+

+(119886 minus 1) 119890


2+

= 119892 (119890119910 119886minus1

)

(5)




11986031

1 119906 = V = 119908 = 119889V

119889119906=

= 119891(119906 V)

= 119892(119906 V)

2 119906 = 119905 V = minus 1198911015840(119905) 119908 =

119889V

119889119906= minus 119891

1015840(119905) minus 119891

10158401015840(119905)

= ℎ(119906 V) minus 1198911015840(119905) minus 119891

10158401015840(119905) = 119896(119906 V)

3 119906 = 119905 V = 119910 119908 = 119889V

119889119906= and = 0

lowast = 119891(119906 V119889V

119889119906) = 0

11986032

1 119906 = 119890119910 V =

119908 =

2

119889V

119889119906=

119890119910(2+ )

minus

1198901199102(2+ )

2

= 119891(119906 V)

119890119910(2+ )

minus

1198901199102(2+ )

= 119892(119906 V)

2 119906 = 119910 V = 119890

119908 =

2

119889V

119889119906= 119890

+ 119890

( minus )

2

2

= 119891(119906 V) 119890

+ 119890

( minus )

2

= 119892(119906 V)

3 119906 = 119910 V =

119908 =

2

119889V

119889119906=

2minus

3

2

= 119891(119906 V)

2minus

3

= 119892(119906 V)

4 119906 = 119910 V = 119905 119908 = 119905119889V

119889119906= 1 +

119905

119905 = 119891(119906 V) 1 +119905

= 119892(119906 V)

11986033

1 119906 = 119910 V =

119908 =

3

119889V

119889119906=

2minus

3

3

= 119891(119906 V)

2minus

3

= 119892(119906 V)

2 119906 = 119910 V =2

22minus

1

119908 =

3

minus

2

119889V

119889119906=

3

+( minus

2)

4

3

minus

2

= 119891(119906 V) 3

+( minus

2)

4

= 119892(119906 V)

3 119906 = 119910 minus1

V =

119908 =

3

119889V

119889119906=

2

( + 2)minus

3

2( +

2)

3

= 119891(119906 V) 2

( + 2)minus

3

2( +

2)= 119892(119906 V)

11986034

1 119906 = 119910 V = 1198901

119908 =

2

119889V

119889119906= 1198901

minus 1198901

(1 + )

3

2

= 119891 (119906 V) 1198901

minus 1198901

(1 + )

3

= 119892(119906 V)

2 119906 = 119910minus1

V = 119890

1

119908 =

2

119889V

119889119906= 1198901

+ 2

minus1198901

(1 + )

( + 2)

2

= 119891(119906 V) 1198901

+ 2

minus 1198901

(1 + )

( + 2)

= 119892(119906 V)

3 119906 = 119910 V = 119908 = 119890119905

119889V

119889119906=

119890119905 = 119891 (119906 V)

= 119892(119906 V)

11986035

1 119906 = 119910 V = 119908 =

2

119889V

119889119906=

2

= 119891(119906 V)

= 119892(119906 V)


Table 2 Continued


2 119906 = V = 119890119910 119908 =

119889V

119889119906= 1198901199102

+ 119890119910

= 119891(119906 V) 119890119910

2

+ 119890119910

= 119892(119906 V)

3 119906 = 119905 V = 119910119889V

119889119906=

1198892V

1198891199062= and = 0

lowast = 0 = 119892(119906 V119889V

119889119906)

4 119906 = 119905 V = 119908 = 119890minus119910

119889V

119889119906=

119890minus119910

= 119891(119906 V) = 119892(119906 V)

119860119886

36

|119886| le 1119886 = 0 1

1 119906 = 119910 V = 119886minus1

119908 =

119889V

119889119906= 119886minus2

+ (119886 minus 1)119886minus3

= 119891(119906 V) 119886minus2 + (119886 minus 1)

119886minus3 = 119892(119906 V)

2 119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906=

119890minus119910

119886minus1

2+

+(119886 minus 1)119890


2+

= 119891(119906 V)

119890minus119910

119886minus1

2+

+(119886 minus 1)119890


2+

= 119892(119906 V)

3 119906 = 119910 V = 119905 119908 = 119905(2119886minus1)(119886minus1)

119889V

119889119906= 1 +

119905

119905(2119886minus1)(119886minus1)

= 119891(119906 V) 1 +119905

= 119892(119906 V)

11986037

1 119906 = 119910 V =119890minus119886 arctan

radic1 + 2

119908 =119890minus119886 arctan

(1 + 2)32

119889V

119889119906=


(1 + 2)32


radic1 + 2


(1 + 2)32


(1 + 2)32


radic1 + 2

= 119892(119906 V)


radic1 + 2

119908 =119890minus119886 arctan

(1 + 2)32

119889V

119889119906=


( + (1 + 2))radic1 +

2

+radic1 +


+ (1 + 2)


(1 + 2)32

= 119891(119906 V)minus(119886 + )119890

minus119886 arctan

( + (1 + 2))radic1 +

2

+radic1 +


+ (1 + 2)

= 119892(119906 V)

3 119906 = 119910 V = (1 + 1199052) 119908 =

119890119886 arctan 119905

32

119889V

119889119906= 2119905 +

(1 + 1199052)

119890119886 arctan 119905

32

= 119891(119906 V) 2119905 +(1 + 119905

2)

= 119892(119906 V)

11986038


119889V

119889119906= minus2 minus

4119909

2 minus


2 minus = 119892(119906 V)

3 119906 = 119910 V =

(119905 minus 119909)22 119908 =

2

(119905 minus 119909)+

2

119889V

119889119906=

(119905 minus 119909)23minus

2(1 minus ) + 2(119905 minus 119909)

(119905 minus 119909)34

2

(119905 minus 119909)+

2

= 119891(119906 V)

(119905 minus 119909)23minus

2(1 minus ) + 2(119905 minus 119909)

(119905 minus 119909)34

= 119892(119906 V)

4 119906 = 119910 V =119905 minus 119909

119908 =

3

119889V

119889119906=

119905

2minus

(119905 minus 119909)

3

3

= 119891(119906 V) 119905

2minus

(119905 minus 119909)

3

= 119892(119906 V)

5 119906 = 119910 V =119909

radic1 + 2

119908 =1 + 2

11990933

+

11990923

119889V

119889119906=

(1 + 2minus 119909)

(1 + 2)32

+119909

radic1 + 2

1 + 2

11990933

+

11990923

= 119891(119906 V) (1 + 2minus 119909)

(1 + 2)32

+119909

radic1 + 2

= 119892(119906 V)


Table 2 Continued


11986039

1 119906 = 119910 V =1 + 2+ (119905 minus 119909)

2

(1 + 1199052 + 1199092)22

119908 =

(1 + 1199052 + 1199092)32

3

119889V

119889119906= minus

2(1 + 2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2

(1 + 1199052 + 1199092)32

3

= 119891 (119906 V)

minus2(1 +

2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2= 119892(119906 V)



119908 = sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

119889V

119889119906=

(cos2119909 + 2) sec3119909


2sec2119909

+((1 +



2sec2119909 sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

= 119891(119906 V)

(cos2119909 + 2) sec3119909


2sec2119909

+((1 +



2sec2119909= 119892(119906 V)


|119886| le


119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906=

119890minus119910

119886minus1

2+

+(119886 minus 1) 119890


2+

(6)






119905minus 119863119905119883 = minus(119863

119905120585)119863119905(see eg [17])


119889119906=

119863119905V

119863119905119906

= 119863V (7)


119863 = (119863119905119906)minus1

119863119905 (8)



1198862

36 |119886| le


1198862

36 |119886| le


119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906= 119863V

119889119908

119889119906= 119863119908

1198892V

1198891199062= 1198632V

(9)


where119863 = 119890minus119910

(2+2)minus1

119863119905and119863119908 and119863


119890minus119910

119909 minus 119890minus119910

( + )

22(2+ )

= 119891 (119906 V 119908 119863V)

(119890minus2119910

119886minus3

(minus5+ (2119886 minus 5)


4

+2 (119886 minus 1) 2)) ((

2+ )3

)minus1

+ (119890minus2119910

119886minus3


3)

+2 119909 (2+)+


119910))

times ((2+ )3

)minus1

= 119892 (119906 V 119908 119863V)

(10)


1198862

36 |119886| le





1198862



3

31 1198603

35 1198604

31 and 119860

1

38in which two

of them 1198603

31and 119860

3


two 1198604

31and 119860

1


system


and1198603

35 we have


Consider 119860331


singular invariants

119906 = 119905 V = 119910 119908 = 119863V = = 0 (11)


= 119891 (119906 V 119863V) = 0 (12)


119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909 minus

119910 119863

3V =

119910

(13)


119909 =

119910

+

1

119891 (119906 V 119863V 119863

2V)

119910 = 119892 (119906 V 119863V 119863

2V)

(14)


3V where 119863 = 119863119905 For



119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909

119863

3V =

119910

(15)


31and

1198601


119906 = 119909 V = 119910 119863V =

1198632V =

2minus

3

(16)


minus1



=

+ 2119892 (119906 V 119863V) (17)


4


invariants

119908 =

11990912060110158401015840(119909)

4

minus3212060110158401015840(119909) +

2120601101584010158401015840

(119909)

5

1198633V =

119910

3+

(32minus

119909) minus 3

5

(18)

and for 119860138


119908 =

119909

3minus

32

24 119863

3V =

119910

3+

(32minus

119909) minus 3

5

(19)






4

31and

1198601






(a) 119906 V 119908119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus2

119908

119889119906119896minus2119889119896minus1V

119889119906119896minus1

(b) 119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus3

119908

119889119906119896minus3

119889119896V

119889119906119896

(20)


119896minus2119908119889119906119896minus2


119896minus3119908119889119906119896minus3

119889119896V119889119906119896


31 1198603

35 1198604

31 and 119860

1

38and (a) to

the remainder

3 Integrability



1198603

31⋍ 120597119909 119905120597119909 119910120597119909 = 119891 (119905 119910 ) = 0 (21)

Type II

1198603

35⋍ 120597119909 119905120597119909 119909120597119909 = 119892 (119905 119910 ) = 0 (22)

Type III

1198604

32⋍ 120597119909 119905120597119905+ 119909120597119909 119905120597119909

119905 = 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198603

34⋍ 120597119909 119905120597119909 minus120597119905+ 119909120597119909

119890119905 = 119891 (119910 )

= 119892 (119910 )

1198604

35⋍ 120597119909 119905120597119909 119909120597119909+ 120597119910

119890minus119910

= 119891 (119905 ) = 119892 (119905 )

1198601198863

36⋍ 120597119909 119905120597119909 (1 minus 119886) 119905120597

119905+ 119909120597119909

119905(2119886minus1)(119886minus1)

= 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198601198863

37⋍ 120597119909 119905120597119909 minus (1 + 119905

2) 120597119905+ (119886 minus 119905) 119909120597

119909

minus32

119890119886 tan 119905

= 119891 (119910 (1 + 1199052))

2119905 +(1 + 119905

2)

= 119892 (119910 (1 + 119905

2))

(23)










= 4 + 3119910 + 119910

3= 0 (24)






31or 11986035



32 1198603

34 1198604

35 119860119886336 and 119860

1198863







= 119890minus11990511991034 =

2119910minus1

minus 1199102 (25)


= 120597119909 1198832

=





119910 By


119905 = 119910 119909 = 119909 119910 = 119905 (26)


1199052

3




1

2sech21198881(119905 minus 119888

2)



119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

times tanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(27)



3





3

35












1 1198882 and 119888

3as well as 119888


constants

321 Uncoupled Cases

1198604

32 Let V = 120582(119906 119888


Table 2 we have

119905 = 120582 (119910 1198881) (28)


1198601

33 Let V = 120582(119906 119888



= 120582 (119910 1198881) (29)


=minus1

int 120582 (119910 1198881) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2

(30)


120582 (119910 1198881) int120582 (119910 119888

1) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2 = minus1 (31)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (32)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (33)


1

33

1198602

33 Let V = 120582(119906 119888



= plusmnradic2 + 22120582 (119910 119888

1) (34)


= minus int119891 (119910 120582 (119910 1198881)) 119889119910 + 119888

2 (35)


=2

(int119891(119910 120582(119910 1198881))119889119910 + 119888

2)2

minus 2120582 (119910 1198881)

(36)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (37)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (38)


2

33

1198603

34 Let V = 120582(119906 119888


= 120582 (119910 1198881) (39)

which implies that

119910 = 120601 (119905 1198881 1198882) (40)


119909 = intint 119890minus119905119891 (120601 120601

119905) 119889119905 119889119905 + 119888

3119905 + 1198884 (41)

which results in

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (42)


3

34

1198601

35 Let V = 120582(119906 119888


we obtain

= 120582 (119910 1198881) (43)


int119889119910

1198882exp int119891 (119910 120582 (119906 119888

1)) 119889119910

= 119905 + 1198883 (44)

Inverting we have

119910 = 120601 (119905 1198881 1198882 1198883) (45)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (46)


1

35

For119860435

1198601198863

36 and119860

1198863



34


322 Coupled Cases

1198601

31 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 ( 1198881) (47)


119909 = 120601 (119905 1198881 1198882 1198883) (48)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (49)


1

31

1198602


1198601

31

1198601

32 Let V = 120582(119906 119888



= 120582 (119890119910 1198881) (50)


= exp int ((120582 (119890119910 1198881) + 119890

119910119892 (119890119910 120582 (119890

119910 1198881)))

times119891 (119890119910 120582 (119890

119910 1198881)) 119889 (119890

119910))

times (119890119910(1+120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881))))minus1

(51)


119889 (119890119910)

119889119905

=2119890119910120582 (119890119910 1198881) (1 + 120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881)))

(120582 (119890119910 1198881) + 119890119910119892 (119890119910 120582 (119890119910 119888

1)))

(52)


119890119910= 120601 (119910 + 119888

2) (53)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (54)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (55)


1

32

1198602


35

1198603


1

33

1198603

33 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 (119910 minus

1

1198881) (56)


119891(119906 V) we arrive at

119910 = 120601(119910 minus1

1198881 1198882) (57)


=1

119910 minus 120601minus1 (119910 1198881 1198882) (58)


(119910 minus 120601minus1

(119910 1198881 1198882)) 120582 (120601

minus1(119910 1198881 1198882) 1198881) = 1 (59)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (60)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (61)


3

33

1198601


1

33

1198602


33

1198602

35 Let V = 120582(119906 119888



119890119910= 120582 ( 119888

1) (62)


= 120601 (119910 1198881 1198882) (63)


119910 = 120595 (119909 1198881 1198882 1198883) (64)


119909 = ℎ (119905 1198881 1198882 1198883 1198884) (65)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (66)


2

35


1198601198861


33

1198601198862


1198601

32

1198601198861


1198601


1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)


= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)


= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)


119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37




(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)




= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603



119906 = 119910 V = (78)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)



minus 119906) (80)


where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11986031

1 120597119905 120597119909 120597119910

3

2 120597119909 120597119910 119905120597119909+ 119891(119905)120597

119910 2

3 120597119909 119905120597119909 119910120597119909

1

4 120597119905 119909120597119905 120601(119909)120597

119905 12060110158401015840(119909) = 0 1

11986032

[1198831 1198832] = 119883

1

1 120597119905 119905120597119905+ 120597119910 120597119909

3

2 120597119905 119905120597119905+ 119910120597119909 120597119909

2

3 120597119905 119905120597119905 120597119909

2

4 120597119909 119905120597119905+ 119909120597119909 119905120597119909

2

11986033

[1198832 1198833] = 119883

1

1 120597119905 120597119909 119909120597119905

2

2 120597119905 120597119909 119909120597119905+ 119910120597119909

2

3 120597119905 120597119909 119909120597119905+ 120597119910

3

11986034

1 120597119905 120597119909 (119905 + 119909)120597

119905+ 119909120597119909

2

[1198831 1198833] = 119883

12 120597

119905 120597119909 (119905 + 119909)120597

119905+ 119909120597119909+ 120597119910

3

[1198832 1198833] = 119883

1+ 1198832

3 120597119909 119905120597119909 minus120597119905+ 119909120597119909

2

11986035

[1198831 1198833] = 119883

1

[1198832 1198833] = 119883

2

1 120597119905 120597119909 119905120597119905+ 119909120597119909

2

2 120597119905 120597119909 119905120597119905+ 119909120597119909+ 120597119910

3

3 120597119909 119905120597119909 119909120597119909

1

4 120597119909 119905120597119909 119909120597119909+ 120597119910

2

119860119886

36 |119886| le 1 119886 = 0 1 1 120597

119905 120597119909 119905120597119905+ 119886119909120597

1199092

[1198831 1198833] = 119883

12 120597

119905 120597119909 119905120597119905+ 119886119909120597

119909+ 120597119910

3

[1198832 1198833] = 119886119883

23 120597

119909 119905120597119909 (1 minus 119886)119905120597

119905+ 119909120597119909

2

119860119886

37 119886 ge 0 1 120597

119905 120597119909 (119886119905 + 119909)120597

119905+ (119886119909 minus 119905)120597

1199092

[1198831 1198833] = 119886119883

1minus 1198832

2 120597119905 120597119909 (119886119905 + 119909)120597

119905+ (119886119909 minus 119905)120597

119909+ 120597119910

3

[1198832 1198833] = 119883

1+ 1198861198832

3 120597119909 119905120597119909 minus(1 + 119905

2)120597119905+ (119886 minus 119905)119909120597

1199092

11986038

sl(2 119877)

[1198831 1198832] = 119883

1

[1198832 1198833] = 119883

3

[1198833 1198831] = minus2119883

2

1 120597119905 119905120597119905 1199052120597119905

1

2 120597119905 119905120597119905+ 119909120597119909 1199052120597119905+ 2119905119909120597

119909+ 119909120597119910

3

3 120597119905+ 120597119909 119905120597119905+ 119909120597119909 1199052120597119905+ 1199092120597119909

2

4 minus1199051205971199091

2(minus119905120597119905+ 119909120597119909) 119909120597119905

2

5 120597119905 119905120597119905+ 119909120597119909 (1199052minus 1199092)120597119905+ 2119905119909120597

1199092

11986039

so(3)[1198831 1198832] = 119883

3

1 (1 + 1199052)120597119905+ 119909119905120597

119909 119909120597119905minus 119905120597119909 minus119909119905120597

119905minus (1 + 119909

2)120597119909

2

[1198833 1198831] = 119883

2

[1198832 1198833] = 119883

1


119909+ sin 119905 sec119909120597

119910 minus cos 119905 tan119909120597

119905+ sin 119905120597

119909+ cos 119905 sec119909120597

119910 120597119905

3


2

minus22

3

= 119891(119910

2

2minus

3)

minus119910

4

+

1199092minus 3 + 3

2

5

= 119892(119910

2

2minus

3)

(3)


1199081= 119910 119908

2=

119908

3=

2 119908

4=

2minus

3

1199085=

119909

2

minus22

3 119908

6= minus

119910

4

+

1199092minus 3 + 3

2

5

(4)


5and 119908




= 119891 (119890

119910 119886minus1

)

119890minus119910

119886minus1

2+

+(119886 minus 1) 119890


2+

= 119892 (119890119910 119886minus1

)

(5)




11986031

1 119906 = V = 119908 = 119889V

119889119906=

= 119891(119906 V)

= 119892(119906 V)

2 119906 = 119905 V = minus 1198911015840(119905) 119908 =

119889V

119889119906= minus 119891

1015840(119905) minus 119891

10158401015840(119905)

= ℎ(119906 V) minus 1198911015840(119905) minus 119891

10158401015840(119905) = 119896(119906 V)

3 119906 = 119905 V = 119910 119908 = 119889V

119889119906= and = 0

lowast = 119891(119906 V119889V

119889119906) = 0

11986032

1 119906 = 119890119910 V =

119908 =

2

119889V

119889119906=

119890119910(2+ )

minus

1198901199102(2+ )

2

= 119891(119906 V)

119890119910(2+ )

minus

1198901199102(2+ )

= 119892(119906 V)

2 119906 = 119910 V = 119890

119908 =

2

119889V

119889119906= 119890

+ 119890

( minus )

2

2

= 119891(119906 V) 119890

+ 119890

( minus )

2

= 119892(119906 V)

3 119906 = 119910 V =

119908 =

2

119889V

119889119906=

2minus

3

2

= 119891(119906 V)

2minus

3

= 119892(119906 V)

4 119906 = 119910 V = 119905 119908 = 119905119889V

119889119906= 1 +

119905

119905 = 119891(119906 V) 1 +119905

= 119892(119906 V)

11986033

1 119906 = 119910 V =

119908 =

3

119889V

119889119906=

2minus

3

3

= 119891(119906 V)

2minus

3

= 119892(119906 V)

2 119906 = 119910 V =2

22minus

1

119908 =

3

minus

2

119889V

119889119906=

3

+( minus

2)

4

3

minus

2

= 119891(119906 V) 3

+( minus

2)

4

= 119892(119906 V)

3 119906 = 119910 minus1

V =

119908 =

3

119889V

119889119906=

2

( + 2)minus

3

2( +

2)

3

= 119891(119906 V) 2

( + 2)minus

3

2( +

2)= 119892(119906 V)

11986034

1 119906 = 119910 V = 1198901

119908 =

2

119889V

119889119906= 1198901

minus 1198901

(1 + )

3

2

= 119891 (119906 V) 1198901

minus 1198901

(1 + )

3

= 119892(119906 V)

2 119906 = 119910minus1

V = 119890

1

119908 =

2

119889V

119889119906= 1198901

+ 2

minus1198901

(1 + )

( + 2)

2

= 119891(119906 V) 1198901

+ 2

minus 1198901

(1 + )

( + 2)

= 119892(119906 V)

3 119906 = 119910 V = 119908 = 119890119905

119889V

119889119906=

119890119905 = 119891 (119906 V)

= 119892(119906 V)

11986035

1 119906 = 119910 V = 119908 =

2

119889V

119889119906=

2

= 119891(119906 V)

= 119892(119906 V)


Table 2 Continued


2 119906 = V = 119890119910 119908 =

119889V

119889119906= 1198901199102

+ 119890119910

= 119891(119906 V) 119890119910

2

+ 119890119910

= 119892(119906 V)

3 119906 = 119905 V = 119910119889V

119889119906=

1198892V

1198891199062= and = 0

lowast = 0 = 119892(119906 V119889V

119889119906)

4 119906 = 119905 V = 119908 = 119890minus119910

119889V

119889119906=

119890minus119910

= 119891(119906 V) = 119892(119906 V)

119860119886

36

|119886| le 1119886 = 0 1

1 119906 = 119910 V = 119886minus1

119908 =

119889V

119889119906= 119886minus2

+ (119886 minus 1)119886minus3

= 119891(119906 V) 119886minus2 + (119886 minus 1)

119886minus3 = 119892(119906 V)

2 119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906=

119890minus119910

119886minus1

2+

+(119886 minus 1)119890


2+

= 119891(119906 V)

119890minus119910

119886minus1

2+

+(119886 minus 1)119890


2+

= 119892(119906 V)

3 119906 = 119910 V = 119905 119908 = 119905(2119886minus1)(119886minus1)

119889V

119889119906= 1 +

119905

119905(2119886minus1)(119886minus1)

= 119891(119906 V) 1 +119905

= 119892(119906 V)

11986037

1 119906 = 119910 V =119890minus119886 arctan

radic1 + 2

119908 =119890minus119886 arctan

(1 + 2)32

119889V

119889119906=


(1 + 2)32


radic1 + 2


(1 + 2)32


(1 + 2)32


radic1 + 2

= 119892(119906 V)


radic1 + 2

119908 =119890minus119886 arctan

(1 + 2)32

119889V

119889119906=


( + (1 + 2))radic1 +

2

+radic1 +


+ (1 + 2)


(1 + 2)32

= 119891(119906 V)minus(119886 + )119890

minus119886 arctan

( + (1 + 2))radic1 +

2

+radic1 +


+ (1 + 2)

= 119892(119906 V)

3 119906 = 119910 V = (1 + 1199052) 119908 =

119890119886 arctan 119905

32

119889V

119889119906= 2119905 +

(1 + 1199052)

119890119886 arctan 119905

32

= 119891(119906 V) 2119905 +(1 + 119905

2)

= 119892(119906 V)

11986038


119889V

119889119906= minus2 minus

4119909

2 minus


2 minus = 119892(119906 V)

3 119906 = 119910 V =

(119905 minus 119909)22 119908 =

2

(119905 minus 119909)+

2

119889V

119889119906=

(119905 minus 119909)23minus

2(1 minus ) + 2(119905 minus 119909)

(119905 minus 119909)34

2

(119905 minus 119909)+

2

= 119891(119906 V)

(119905 minus 119909)23minus

2(1 minus ) + 2(119905 minus 119909)

(119905 minus 119909)34

= 119892(119906 V)

4 119906 = 119910 V =119905 minus 119909

119908 =

3

119889V

119889119906=

119905

2minus

(119905 minus 119909)

3

3

= 119891(119906 V) 119905

2minus

(119905 minus 119909)

3

= 119892(119906 V)

5 119906 = 119910 V =119909

radic1 + 2

119908 =1 + 2

11990933

+

11990923

119889V

119889119906=

(1 + 2minus 119909)

(1 + 2)32

+119909

radic1 + 2

1 + 2

11990933

+

11990923

= 119891(119906 V) (1 + 2minus 119909)

(1 + 2)32

+119909

radic1 + 2

= 119892(119906 V)


Table 2 Continued


11986039

1 119906 = 119910 V =1 + 2+ (119905 minus 119909)

2

(1 + 1199052 + 1199092)22

119908 =

(1 + 1199052 + 1199092)32

3

119889V

119889119906= minus

2(1 + 2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2

(1 + 1199052 + 1199092)32

3

= 119891 (119906 V)

minus2(1 +

2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2= 119892(119906 V)



119908 = sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

119889V

119889119906=

(cos2119909 + 2) sec3119909


2sec2119909

+((1 +



2sec2119909 sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

= 119891(119906 V)

(cos2119909 + 2) sec3119909


2sec2119909

+((1 +



2sec2119909= 119892(119906 V)


|119886| le


119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906=

119890minus119910

119886minus1

2+

+(119886 minus 1) 119890


2+

(6)






119905minus 119863119905119883 = minus(119863

119905120585)119863119905(see eg [17])


119889119906=

119863119905V

119863119905119906

= 119863V (7)


119863 = (119863119905119906)minus1

119863119905 (8)



1198862

36 |119886| le


1198862

36 |119886| le


119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906= 119863V

119889119908

119889119906= 119863119908

1198892V

1198891199062= 1198632V

(9)


where119863 = 119890minus119910

(2+2)minus1

119863119905and119863119908 and119863


119890minus119910

119909 minus 119890minus119910

( + )

22(2+ )

= 119891 (119906 V 119908 119863V)

(119890minus2119910

119886minus3

(minus5+ (2119886 minus 5)


4

+2 (119886 minus 1) 2)) ((

2+ )3

)minus1

+ (119890minus2119910

119886minus3


3)

+2 119909 (2+)+


119910))

times ((2+ )3

)minus1

= 119892 (119906 V 119908 119863V)

(10)


1198862

36 |119886| le





1198862



3

31 1198603

35 1198604

31 and 119860

1

38in which two

of them 1198603

31and 119860

3


two 1198604

31and 119860

1


system


and1198603

35 we have


Consider 119860331


singular invariants

119906 = 119905 V = 119910 119908 = 119863V = = 0 (11)


= 119891 (119906 V 119863V) = 0 (12)


119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909 minus

119910 119863

3V =

119910

(13)


119909 =

119910

+

1

119891 (119906 V 119863V 119863

2V)

119910 = 119892 (119906 V 119863V 119863

2V)

(14)


3V where 119863 = 119863119905 For



119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909

119863

3V =

119910

(15)


31and

1198601


119906 = 119909 V = 119910 119863V =

1198632V =

2minus

3

(16)


minus1



=

+ 2119892 (119906 V 119863V) (17)


4


invariants

119908 =

11990912060110158401015840(119909)

4

minus3212060110158401015840(119909) +

2120601101584010158401015840

(119909)

5

1198633V =

119910

3+

(32minus

119909) minus 3

5

(18)

and for 119860138


119908 =

119909

3minus

32

24 119863

3V =

119910

3+

(32minus

119909) minus 3

5

(19)






4

31and

1198601






(a) 119906 V 119908119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus2

119908

119889119906119896minus2119889119896minus1V

119889119906119896minus1

(b) 119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus3

119908

119889119906119896minus3

119889119896V

119889119906119896

(20)


119896minus2119908119889119906119896minus2


119896minus3119908119889119906119896minus3

119889119896V119889119906119896


31 1198603

35 1198604

31 and 119860

1

38and (a) to

the remainder

3 Integrability



1198603

31⋍ 120597119909 119905120597119909 119910120597119909 = 119891 (119905 119910 ) = 0 (21)

Type II

1198603

35⋍ 120597119909 119905120597119909 119909120597119909 = 119892 (119905 119910 ) = 0 (22)

Type III

1198604

32⋍ 120597119909 119905120597119905+ 119909120597119909 119905120597119909

119905 = 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198603

34⋍ 120597119909 119905120597119909 minus120597119905+ 119909120597119909

119890119905 = 119891 (119910 )

= 119892 (119910 )

1198604

35⋍ 120597119909 119905120597119909 119909120597119909+ 120597119910

119890minus119910

= 119891 (119905 ) = 119892 (119905 )

1198601198863

36⋍ 120597119909 119905120597119909 (1 minus 119886) 119905120597

119905+ 119909120597119909

119905(2119886minus1)(119886minus1)

= 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198601198863

37⋍ 120597119909 119905120597119909 minus (1 + 119905

2) 120597119905+ (119886 minus 119905) 119909120597

119909

minus32

119890119886 tan 119905

= 119891 (119910 (1 + 1199052))

2119905 +(1 + 119905

2)

= 119892 (119910 (1 + 119905

2))

(23)










= 4 + 3119910 + 119910

3= 0 (24)






31or 11986035



32 1198603

34 1198604

35 119860119886336 and 119860

1198863







= 119890minus11990511991034 =

2119910minus1

minus 1199102 (25)


= 120597119909 1198832

=





119910 By


119905 = 119910 119909 = 119909 119910 = 119905 (26)


1199052

3




1

2sech21198881(119905 minus 119888

2)



119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

times tanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(27)



3





3

35












1 1198882 and 119888

3as well as 119888


constants

321 Uncoupled Cases

1198604

32 Let V = 120582(119906 119888


Table 2 we have

119905 = 120582 (119910 1198881) (28)


1198601

33 Let V = 120582(119906 119888



= 120582 (119910 1198881) (29)


=minus1

int 120582 (119910 1198881) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2

(30)


120582 (119910 1198881) int120582 (119910 119888

1) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2 = minus1 (31)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (32)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (33)


1

33

1198602

33 Let V = 120582(119906 119888



= plusmnradic2 + 22120582 (119910 119888

1) (34)


= minus int119891 (119910 120582 (119910 1198881)) 119889119910 + 119888

2 (35)


=2

(int119891(119910 120582(119910 1198881))119889119910 + 119888

2)2

minus 2120582 (119910 1198881)

(36)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (37)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (38)


2

33

1198603

34 Let V = 120582(119906 119888


= 120582 (119910 1198881) (39)

which implies that

119910 = 120601 (119905 1198881 1198882) (40)


119909 = intint 119890minus119905119891 (120601 120601

119905) 119889119905 119889119905 + 119888

3119905 + 1198884 (41)

which results in

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (42)


3

34

1198601

35 Let V = 120582(119906 119888


we obtain

= 120582 (119910 1198881) (43)


int119889119910

1198882exp int119891 (119910 120582 (119906 119888

1)) 119889119910

= 119905 + 1198883 (44)

Inverting we have

119910 = 120601 (119905 1198881 1198882 1198883) (45)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (46)


1

35

For119860435

1198601198863

36 and119860

1198863



34


322 Coupled Cases

1198601

31 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 ( 1198881) (47)


119909 = 120601 (119905 1198881 1198882 1198883) (48)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (49)


1

31

1198602


1198601

31

1198601

32 Let V = 120582(119906 119888



= 120582 (119890119910 1198881) (50)


= exp int ((120582 (119890119910 1198881) + 119890

119910119892 (119890119910 120582 (119890

119910 1198881)))

times119891 (119890119910 120582 (119890

119910 1198881)) 119889 (119890

119910))

times (119890119910(1+120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881))))minus1

(51)


119889 (119890119910)

119889119905

=2119890119910120582 (119890119910 1198881) (1 + 120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881)))

(120582 (119890119910 1198881) + 119890119910119892 (119890119910 120582 (119890119910 119888

1)))

(52)


119890119910= 120601 (119910 + 119888

2) (53)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (54)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (55)


1

32

1198602


35

1198603


1

33

1198603

33 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 (119910 minus

1

1198881) (56)


119891(119906 V) we arrive at

119910 = 120601(119910 minus1

1198881 1198882) (57)


=1

119910 minus 120601minus1 (119910 1198881 1198882) (58)


(119910 minus 120601minus1

(119910 1198881 1198882)) 120582 (120601

minus1(119910 1198881 1198882) 1198881) = 1 (59)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (60)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (61)


3

33

1198601


1

33

1198602


33

1198602

35 Let V = 120582(119906 119888



119890119910= 120582 ( 119888

1) (62)


= 120601 (119910 1198881 1198882) (63)


119910 = 120595 (119909 1198881 1198882 1198883) (64)


119909 = ℎ (119905 1198881 1198882 1198883 1198884) (65)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (66)


2

35


1198601198861


33

1198601198862


1198601

32

1198601198861


1198601


1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)


= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)


= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)


119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37




(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)




= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603



119906 = 119910 V = (78)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)



minus 119906) (80)


where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11986031

1 119906 = V = 119908 = 119889V

119889119906=

= 119891(119906 V)

= 119892(119906 V)

2 119906 = 119905 V = minus 1198911015840(119905) 119908 =

119889V

119889119906= minus 119891

1015840(119905) minus 119891

10158401015840(119905)

= ℎ(119906 V) minus 1198911015840(119905) minus 119891

10158401015840(119905) = 119896(119906 V)

3 119906 = 119905 V = 119910 119908 = 119889V

119889119906= and = 0

lowast = 119891(119906 V119889V

119889119906) = 0

11986032

1 119906 = 119890119910 V =

119908 =

2

119889V

119889119906=

119890119910(2+ )

minus

1198901199102(2+ )

2

= 119891(119906 V)

119890119910(2+ )

minus

1198901199102(2+ )

= 119892(119906 V)

2 119906 = 119910 V = 119890

119908 =

2

119889V

119889119906= 119890

+ 119890

( minus )

2

2

= 119891(119906 V) 119890

+ 119890

( minus )

2

= 119892(119906 V)

3 119906 = 119910 V =

119908 =

2

119889V

119889119906=

2minus

3

2

= 119891(119906 V)

2minus

3

= 119892(119906 V)

4 119906 = 119910 V = 119905 119908 = 119905119889V

119889119906= 1 +

119905

119905 = 119891(119906 V) 1 +119905

= 119892(119906 V)

11986033

1 119906 = 119910 V =

119908 =

3

119889V

119889119906=

2minus

3

3

= 119891(119906 V)

2minus

3

= 119892(119906 V)

2 119906 = 119910 V =2

22minus

1

119908 =

3

minus

2

119889V

119889119906=

3

+( minus

2)

4

3

minus

2

= 119891(119906 V) 3

+( minus

2)

4

= 119892(119906 V)

3 119906 = 119910 minus1

V =

119908 =

3

119889V

119889119906=

2

( + 2)minus

3

2( +

2)

3

= 119891(119906 V) 2

( + 2)minus

3

2( +

2)= 119892(119906 V)

11986034

1 119906 = 119910 V = 1198901

119908 =

2

119889V

119889119906= 1198901

minus 1198901

(1 + )

3

2

= 119891 (119906 V) 1198901

minus 1198901

(1 + )

3

= 119892(119906 V)

2 119906 = 119910minus1

V = 119890

1

119908 =

2

119889V

119889119906= 1198901

+ 2

minus1198901

(1 + )

( + 2)

2

= 119891(119906 V) 1198901

+ 2

minus 1198901

(1 + )

( + 2)

= 119892(119906 V)

3 119906 = 119910 V = 119908 = 119890119905

119889V

119889119906=

119890119905 = 119891 (119906 V)

= 119892(119906 V)

11986035

1 119906 = 119910 V = 119908 =

2

119889V

119889119906=

2

= 119891(119906 V)

= 119892(119906 V)


Table 2 Continued


2 119906 = V = 119890119910 119908 =

119889V

119889119906= 1198901199102

+ 119890119910

= 119891(119906 V) 119890119910

2

+ 119890119910

= 119892(119906 V)

3 119906 = 119905 V = 119910119889V

119889119906=

1198892V

1198891199062= and = 0

lowast = 0 = 119892(119906 V119889V

119889119906)

4 119906 = 119905 V = 119908 = 119890minus119910

119889V

119889119906=

119890minus119910

= 119891(119906 V) = 119892(119906 V)

119860119886

36

|119886| le 1119886 = 0 1

1 119906 = 119910 V = 119886minus1

119908 =

119889V

119889119906= 119886minus2

+ (119886 minus 1)119886minus3

= 119891(119906 V) 119886minus2 + (119886 minus 1)

119886minus3 = 119892(119906 V)

2 119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906=

119890minus119910

119886minus1

2+

+(119886 minus 1)119890


2+

= 119891(119906 V)

119890minus119910

119886minus1

2+

+(119886 minus 1)119890


2+

= 119892(119906 V)

3 119906 = 119910 V = 119905 119908 = 119905(2119886minus1)(119886minus1)

119889V

119889119906= 1 +

119905

119905(2119886minus1)(119886minus1)

= 119891(119906 V) 1 +119905

= 119892(119906 V)

11986037

1 119906 = 119910 V =119890minus119886 arctan

radic1 + 2

119908 =119890minus119886 arctan

(1 + 2)32

119889V

119889119906=


(1 + 2)32


radic1 + 2


(1 + 2)32


(1 + 2)32


radic1 + 2

= 119892(119906 V)


radic1 + 2

119908 =119890minus119886 arctan

(1 + 2)32

119889V

119889119906=


( + (1 + 2))radic1 +

2

+radic1 +


+ (1 + 2)


(1 + 2)32

= 119891(119906 V)minus(119886 + )119890

minus119886 arctan

( + (1 + 2))radic1 +

2

+radic1 +


+ (1 + 2)

= 119892(119906 V)

3 119906 = 119910 V = (1 + 1199052) 119908 =

119890119886 arctan 119905

32

119889V

119889119906= 2119905 +

(1 + 1199052)

119890119886 arctan 119905

32

= 119891(119906 V) 2119905 +(1 + 119905

2)

= 119892(119906 V)

11986038


119889V

119889119906= minus2 minus

4119909

2 minus


2 minus = 119892(119906 V)

3 119906 = 119910 V =

(119905 minus 119909)22 119908 =

2

(119905 minus 119909)+

2

119889V

119889119906=

(119905 minus 119909)23minus

2(1 minus ) + 2(119905 minus 119909)

(119905 minus 119909)34

2

(119905 minus 119909)+

2

= 119891(119906 V)

(119905 minus 119909)23minus

2(1 minus ) + 2(119905 minus 119909)

(119905 minus 119909)34

= 119892(119906 V)

4 119906 = 119910 V =119905 minus 119909

119908 =

3

119889V

119889119906=

119905

2minus

(119905 minus 119909)

3

3

= 119891(119906 V) 119905

2minus

(119905 minus 119909)

3

= 119892(119906 V)

5 119906 = 119910 V =119909

radic1 + 2

119908 =1 + 2

11990933

+

11990923

119889V

119889119906=

(1 + 2minus 119909)

(1 + 2)32

+119909

radic1 + 2

1 + 2

11990933

+

11990923

= 119891(119906 V) (1 + 2minus 119909)

(1 + 2)32

+119909

radic1 + 2

= 119892(119906 V)


Table 2 Continued


11986039

1 119906 = 119910 V =1 + 2+ (119905 minus 119909)

2

(1 + 1199052 + 1199092)22

119908 =

(1 + 1199052 + 1199092)32

3

119889V

119889119906= minus

2(1 + 2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2

(1 + 1199052 + 1199092)32

3

= 119891 (119906 V)

minus2(1 +

2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2= 119892(119906 V)



119908 = sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

119889V

119889119906=

(cos2119909 + 2) sec3119909


2sec2119909

+((1 +



2sec2119909 sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

= 119891(119906 V)

(cos2119909 + 2) sec3119909


2sec2119909

+((1 +



2sec2119909= 119892(119906 V)


|119886| le


119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906=

119890minus119910

119886minus1

2+

+(119886 minus 1) 119890


2+

(6)






119905minus 119863119905119883 = minus(119863

119905120585)119863119905(see eg [17])


119889119906=

119863119905V

119863119905119906

= 119863V (7)


119863 = (119863119905119906)minus1

119863119905 (8)



1198862

36 |119886| le


1198862

36 |119886| le


119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906= 119863V

119889119908

119889119906= 119863119908

1198892V

1198891199062= 1198632V

(9)


where119863 = 119890minus119910

(2+2)minus1

119863119905and119863119908 and119863


119890minus119910

119909 minus 119890minus119910

( + )

22(2+ )

= 119891 (119906 V 119908 119863V)

(119890minus2119910

119886minus3

(minus5+ (2119886 minus 5)


4

+2 (119886 minus 1) 2)) ((

2+ )3

)minus1

+ (119890minus2119910

119886minus3


3)

+2 119909 (2+)+


119910))

times ((2+ )3

)minus1

= 119892 (119906 V 119908 119863V)

(10)


1198862

36 |119886| le





1198862



3

31 1198603

35 1198604

31 and 119860

1

38in which two

of them 1198603

31and 119860

3


two 1198604

31and 119860

1


system


and1198603

35 we have


Consider 119860331


singular invariants

119906 = 119905 V = 119910 119908 = 119863V = = 0 (11)


= 119891 (119906 V 119863V) = 0 (12)


119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909 minus

119910 119863

3V =

119910

(13)


119909 =

119910

+

1

119891 (119906 V 119863V 119863

2V)

119910 = 119892 (119906 V 119863V 119863

2V)

(14)


3V where 119863 = 119863119905 For



119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909

119863

3V =

119910

(15)


31and

1198601


119906 = 119909 V = 119910 119863V =

1198632V =

2minus

3

(16)


minus1



=

+ 2119892 (119906 V 119863V) (17)


4


invariants

119908 =

11990912060110158401015840(119909)

4

minus3212060110158401015840(119909) +

2120601101584010158401015840

(119909)

5

1198633V =

119910

3+

(32minus

119909) minus 3

5

(18)

and for 119860138


119908 =

119909

3minus

32

24 119863

3V =

119910

3+

(32minus

119909) minus 3

5

(19)






4

31and

1198601






(a) 119906 V 119908119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus2

119908

119889119906119896minus2119889119896minus1V

119889119906119896minus1

(b) 119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus3

119908

119889119906119896minus3

119889119896V

119889119906119896

(20)


119896minus2119908119889119906119896minus2


119896minus3119908119889119906119896minus3

119889119896V119889119906119896


31 1198603

35 1198604

31 and 119860

1

38and (a) to

the remainder

3 Integrability



1198603

31⋍ 120597119909 119905120597119909 119910120597119909 = 119891 (119905 119910 ) = 0 (21)

Type II

1198603

35⋍ 120597119909 119905120597119909 119909120597119909 = 119892 (119905 119910 ) = 0 (22)

Type III

1198604

32⋍ 120597119909 119905120597119905+ 119909120597119909 119905120597119909

119905 = 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198603

34⋍ 120597119909 119905120597119909 minus120597119905+ 119909120597119909

119890119905 = 119891 (119910 )

= 119892 (119910 )

1198604

35⋍ 120597119909 119905120597119909 119909120597119909+ 120597119910

119890minus119910

= 119891 (119905 ) = 119892 (119905 )

1198601198863

36⋍ 120597119909 119905120597119909 (1 minus 119886) 119905120597

119905+ 119909120597119909

119905(2119886minus1)(119886minus1)

= 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198601198863

37⋍ 120597119909 119905120597119909 minus (1 + 119905

2) 120597119905+ (119886 minus 119905) 119909120597

119909

minus32

119890119886 tan 119905

= 119891 (119910 (1 + 1199052))

2119905 +(1 + 119905

2)

= 119892 (119910 (1 + 119905

2))

(23)










= 4 + 3119910 + 119910

3= 0 (24)






31or 11986035



32 1198603

34 1198604

35 119860119886336 and 119860

1198863







= 119890minus11990511991034 =

2119910minus1

minus 1199102 (25)


= 120597119909 1198832

=





119910 By


119905 = 119910 119909 = 119909 119910 = 119905 (26)


1199052

3




1

2sech21198881(119905 minus 119888

2)



119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

times tanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(27)



3





3

35












1 1198882 and 119888

3as well as 119888


constants

321 Uncoupled Cases

1198604

32 Let V = 120582(119906 119888


Table 2 we have

119905 = 120582 (119910 1198881) (28)


1198601

33 Let V = 120582(119906 119888



= 120582 (119910 1198881) (29)


=minus1

int 120582 (119910 1198881) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2

(30)


120582 (119910 1198881) int120582 (119910 119888

1) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2 = minus1 (31)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (32)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (33)


1

33

1198602

33 Let V = 120582(119906 119888



= plusmnradic2 + 22120582 (119910 119888

1) (34)


= minus int119891 (119910 120582 (119910 1198881)) 119889119910 + 119888

2 (35)


=2

(int119891(119910 120582(119910 1198881))119889119910 + 119888

2)2

minus 2120582 (119910 1198881)

(36)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (37)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (38)


2

33

1198603

34 Let V = 120582(119906 119888


= 120582 (119910 1198881) (39)

which implies that

119910 = 120601 (119905 1198881 1198882) (40)


119909 = intint 119890minus119905119891 (120601 120601

119905) 119889119905 119889119905 + 119888

3119905 + 1198884 (41)

which results in

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (42)


3

34

1198601

35 Let V = 120582(119906 119888


we obtain

= 120582 (119910 1198881) (43)


int119889119910

1198882exp int119891 (119910 120582 (119906 119888

1)) 119889119910

= 119905 + 1198883 (44)

Inverting we have

119910 = 120601 (119905 1198881 1198882 1198883) (45)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (46)


1

35

For119860435

1198601198863

36 and119860

1198863



34


322 Coupled Cases

1198601

31 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 ( 1198881) (47)


119909 = 120601 (119905 1198881 1198882 1198883) (48)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (49)


1

31

1198602


1198601

31

1198601

32 Let V = 120582(119906 119888



= 120582 (119890119910 1198881) (50)


= exp int ((120582 (119890119910 1198881) + 119890

119910119892 (119890119910 120582 (119890

119910 1198881)))

times119891 (119890119910 120582 (119890

119910 1198881)) 119889 (119890

119910))

times (119890119910(1+120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881))))minus1

(51)


119889 (119890119910)

119889119905

=2119890119910120582 (119890119910 1198881) (1 + 120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881)))

(120582 (119890119910 1198881) + 119890119910119892 (119890119910 120582 (119890119910 119888

1)))

(52)


119890119910= 120601 (119910 + 119888

2) (53)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (54)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (55)


1

32

1198602


35

1198603


1

33

1198603

33 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 (119910 minus

1

1198881) (56)


119891(119906 V) we arrive at

119910 = 120601(119910 minus1

1198881 1198882) (57)


=1

119910 minus 120601minus1 (119910 1198881 1198882) (58)


(119910 minus 120601minus1

(119910 1198881 1198882)) 120582 (120601

minus1(119910 1198881 1198882) 1198881) = 1 (59)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (60)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (61)


3

33

1198601


1

33

1198602


33

1198602

35 Let V = 120582(119906 119888



119890119910= 120582 ( 119888

1) (62)


= 120601 (119910 1198881 1198882) (63)


119910 = 120595 (119909 1198881 1198882 1198883) (64)


119909 = ℎ (119905 1198881 1198882 1198883 1198884) (65)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (66)


2

35


1198601198861


33

1198601198862


1198601

32

1198601198861


1198601


1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)


= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)


= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)


119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37




(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)




= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603



119906 = 119910 V = (78)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)



minus 119906) (80)


where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 2 Continued


2 119906 = V = 119890119910 119908 =

119889V

119889119906= 1198901199102

+ 119890119910

= 119891(119906 V) 119890119910

2

+ 119890119910

= 119892(119906 V)

3 119906 = 119905 V = 119910119889V

119889119906=

1198892V

1198891199062= and = 0

lowast = 0 = 119892(119906 V119889V

119889119906)

4 119906 = 119905 V = 119908 = 119890minus119910

119889V

119889119906=

119890minus119910

= 119891(119906 V) = 119892(119906 V)

119860119886

36

|119886| le 1119886 = 0 1

1 119906 = 119910 V = 119886minus1

119908 =

119889V

119889119906= 119886minus2

+ (119886 minus 1)119886minus3

= 119891(119906 V) 119886minus2 + (119886 minus 1)

119886minus3 = 119892(119906 V)

2 119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906=

119890minus119910

119886minus1

2+

+(119886 minus 1)119890


2+

= 119891(119906 V)

119890minus119910

119886minus1

2+

+(119886 minus 1)119890


2+

= 119892(119906 V)

3 119906 = 119910 V = 119905 119908 = 119905(2119886minus1)(119886minus1)

119889V

119889119906= 1 +

119905

119905(2119886minus1)(119886minus1)

= 119891(119906 V) 1 +119905

= 119892(119906 V)

11986037

1 119906 = 119910 V =119890minus119886 arctan

radic1 + 2

119908 =119890minus119886 arctan

(1 + 2)32

119889V

119889119906=


(1 + 2)32


radic1 + 2


(1 + 2)32


(1 + 2)32


radic1 + 2

= 119892(119906 V)


radic1 + 2

119908 =119890minus119886 arctan

(1 + 2)32

119889V

119889119906=


( + (1 + 2))radic1 +

2

+radic1 +


+ (1 + 2)


(1 + 2)32

= 119891(119906 V)minus(119886 + )119890

minus119886 arctan

( + (1 + 2))radic1 +

2

+radic1 +


+ (1 + 2)

= 119892(119906 V)

3 119906 = 119910 V = (1 + 1199052) 119908 =

119890119886 arctan 119905

32

119889V

119889119906= 2119905 +

(1 + 1199052)

119890119886 arctan 119905

32

= 119891(119906 V) 2119905 +(1 + 119905

2)

= 119892(119906 V)

11986038


119889V

119889119906= minus2 minus

4119909

2 minus


2 minus = 119892(119906 V)

3 119906 = 119910 V =

(119905 minus 119909)22 119908 =

2

(119905 minus 119909)+

2

119889V

119889119906=

(119905 minus 119909)23minus

2(1 minus ) + 2(119905 minus 119909)

(119905 minus 119909)34

2

(119905 minus 119909)+

2

= 119891(119906 V)

(119905 minus 119909)23minus

2(1 minus ) + 2(119905 minus 119909)

(119905 minus 119909)34

= 119892(119906 V)

4 119906 = 119910 V =119905 minus 119909

119908 =

3

119889V

119889119906=

119905

2minus

(119905 minus 119909)

3

3

= 119891(119906 V) 119905

2minus

(119905 minus 119909)

3

= 119892(119906 V)

5 119906 = 119910 V =119909

radic1 + 2

119908 =1 + 2

11990933

+

11990923

119889V

119889119906=

(1 + 2minus 119909)

(1 + 2)32

+119909

radic1 + 2

1 + 2

11990933

+

11990923

= 119891(119906 V) (1 + 2minus 119909)

(1 + 2)32

+119909

radic1 + 2

= 119892(119906 V)


Table 2 Continued


11986039

1 119906 = 119910 V =1 + 2+ (119905 minus 119909)

2

(1 + 1199052 + 1199092)22

119908 =

(1 + 1199052 + 1199092)32

3

119889V

119889119906= minus

2(1 + 2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2

(1 + 1199052 + 1199092)32

3

= 119891 (119906 V)

minus2(1 +

2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2= 119892(119906 V)



119908 = sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

119889V

119889119906=

(cos2119909 + 2) sec3119909


2sec2119909

+((1 +



2sec2119909 sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

= 119891(119906 V)

(cos2119909 + 2) sec3119909


2sec2119909

+((1 +



2sec2119909= 119892(119906 V)


|119886| le


119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906=

119890minus119910

119886minus1

2+

+(119886 minus 1) 119890


2+

(6)






119905minus 119863119905119883 = minus(119863

119905120585)119863119905(see eg [17])


119889119906=

119863119905V

119863119905119906

= 119863V (7)


119863 = (119863119905119906)minus1

119863119905 (8)



1198862

36 |119886| le


1198862

36 |119886| le


119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906= 119863V

119889119908

119889119906= 119863119908

1198892V

1198891199062= 1198632V

(9)


where119863 = 119890minus119910

(2+2)minus1

119863119905and119863119908 and119863


119890minus119910

119909 minus 119890minus119910

( + )

22(2+ )

= 119891 (119906 V 119908 119863V)

(119890minus2119910

119886minus3

(minus5+ (2119886 minus 5)


4

+2 (119886 minus 1) 2)) ((

2+ )3

)minus1

+ (119890minus2119910

119886minus3


3)

+2 119909 (2+)+


119910))

times ((2+ )3

)minus1

= 119892 (119906 V 119908 119863V)

(10)


1198862

36 |119886| le





1198862



3

31 1198603

35 1198604

31 and 119860

1

38in which two

of them 1198603

31and 119860

3


two 1198604

31and 119860

1


system


and1198603

35 we have


Consider 119860331


singular invariants

119906 = 119905 V = 119910 119908 = 119863V = = 0 (11)


= 119891 (119906 V 119863V) = 0 (12)


119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909 minus

119910 119863

3V =

119910

(13)


119909 =

119910

+

1

119891 (119906 V 119863V 119863

2V)

119910 = 119892 (119906 V 119863V 119863

2V)

(14)


3V where 119863 = 119863119905 For



119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909

119863

3V =

119910

(15)


31and

1198601


119906 = 119909 V = 119910 119863V =

1198632V =

2minus

3

(16)


minus1



=

+ 2119892 (119906 V 119863V) (17)


4


invariants

119908 =

11990912060110158401015840(119909)

4

minus3212060110158401015840(119909) +

2120601101584010158401015840

(119909)

5

1198633V =

119910

3+

(32minus

119909) minus 3

5

(18)

and for 119860138


119908 =

119909

3minus

32

24 119863

3V =

119910

3+

(32minus

119909) minus 3

5

(19)






4

31and

1198601






(a) 119906 V 119908119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus2

119908

119889119906119896minus2119889119896minus1V

119889119906119896minus1

(b) 119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus3

119908

119889119906119896minus3

119889119896V

119889119906119896

(20)


119896minus2119908119889119906119896minus2


119896minus3119908119889119906119896minus3

119889119896V119889119906119896


31 1198603

35 1198604

31 and 119860

1

38and (a) to

the remainder

3 Integrability



1198603

31⋍ 120597119909 119905120597119909 119910120597119909 = 119891 (119905 119910 ) = 0 (21)

Type II

1198603

35⋍ 120597119909 119905120597119909 119909120597119909 = 119892 (119905 119910 ) = 0 (22)

Type III

1198604

32⋍ 120597119909 119905120597119905+ 119909120597119909 119905120597119909

119905 = 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198603

34⋍ 120597119909 119905120597119909 minus120597119905+ 119909120597119909

119890119905 = 119891 (119910 )

= 119892 (119910 )

1198604

35⋍ 120597119909 119905120597119909 119909120597119909+ 120597119910

119890minus119910

= 119891 (119905 ) = 119892 (119905 )

1198601198863

36⋍ 120597119909 119905120597119909 (1 minus 119886) 119905120597

119905+ 119909120597119909

119905(2119886minus1)(119886minus1)

= 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198601198863

37⋍ 120597119909 119905120597119909 minus (1 + 119905

2) 120597119905+ (119886 minus 119905) 119909120597

119909

minus32

119890119886 tan 119905

= 119891 (119910 (1 + 1199052))

2119905 +(1 + 119905

2)

= 119892 (119910 (1 + 119905

2))

(23)










= 4 + 3119910 + 119910

3= 0 (24)






31or 11986035



32 1198603

34 1198604

35 119860119886336 and 119860

1198863







= 119890minus11990511991034 =

2119910minus1

minus 1199102 (25)


= 120597119909 1198832

=





119910 By


119905 = 119910 119909 = 119909 119910 = 119905 (26)


1199052

3




1

2sech21198881(119905 minus 119888

2)



119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

times tanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(27)



3





3

35












1 1198882 and 119888

3as well as 119888


constants

321 Uncoupled Cases

1198604

32 Let V = 120582(119906 119888


Table 2 we have

119905 = 120582 (119910 1198881) (28)


1198601

33 Let V = 120582(119906 119888



= 120582 (119910 1198881) (29)


=minus1

int 120582 (119910 1198881) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2

(30)


120582 (119910 1198881) int120582 (119910 119888

1) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2 = minus1 (31)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (32)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (33)


1

33

1198602

33 Let V = 120582(119906 119888



= plusmnradic2 + 22120582 (119910 119888

1) (34)


= minus int119891 (119910 120582 (119910 1198881)) 119889119910 + 119888

2 (35)


=2

(int119891(119910 120582(119910 1198881))119889119910 + 119888

2)2

minus 2120582 (119910 1198881)

(36)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (37)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (38)


2

33

1198603

34 Let V = 120582(119906 119888


= 120582 (119910 1198881) (39)

which implies that

119910 = 120601 (119905 1198881 1198882) (40)


119909 = intint 119890minus119905119891 (120601 120601

119905) 119889119905 119889119905 + 119888

3119905 + 1198884 (41)

which results in

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (42)


3

34

1198601

35 Let V = 120582(119906 119888


we obtain

= 120582 (119910 1198881) (43)


int119889119910

1198882exp int119891 (119910 120582 (119906 119888

1)) 119889119910

= 119905 + 1198883 (44)

Inverting we have

119910 = 120601 (119905 1198881 1198882 1198883) (45)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (46)


1

35

For119860435

1198601198863

36 and119860

1198863



34


322 Coupled Cases

1198601

31 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 ( 1198881) (47)


119909 = 120601 (119905 1198881 1198882 1198883) (48)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (49)


1

31

1198602


1198601

31

1198601

32 Let V = 120582(119906 119888



= 120582 (119890119910 1198881) (50)


= exp int ((120582 (119890119910 1198881) + 119890

119910119892 (119890119910 120582 (119890

119910 1198881)))

times119891 (119890119910 120582 (119890

119910 1198881)) 119889 (119890

119910))

times (119890119910(1+120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881))))minus1

(51)


119889 (119890119910)

119889119905

=2119890119910120582 (119890119910 1198881) (1 + 120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881)))

(120582 (119890119910 1198881) + 119890119910119892 (119890119910 120582 (119890119910 119888

1)))

(52)


119890119910= 120601 (119910 + 119888

2) (53)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (54)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (55)


1

32

1198602


35

1198603


1

33

1198603

33 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 (119910 minus

1

1198881) (56)


119891(119906 V) we arrive at

119910 = 120601(119910 minus1

1198881 1198882) (57)


=1

119910 minus 120601minus1 (119910 1198881 1198882) (58)


(119910 minus 120601minus1

(119910 1198881 1198882)) 120582 (120601

minus1(119910 1198881 1198882) 1198881) = 1 (59)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (60)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (61)


3

33

1198601


1

33

1198602


33

1198602

35 Let V = 120582(119906 119888



119890119910= 120582 ( 119888

1) (62)


= 120601 (119910 1198881 1198882) (63)


119910 = 120595 (119909 1198881 1198882 1198883) (64)


119909 = ℎ (119905 1198881 1198882 1198883 1198884) (65)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (66)


2

35


1198601198861


33

1198601198862


1198601

32

1198601198861


1198601


1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)


= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)


= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)


119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37




(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)




= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603



119906 = 119910 V = (78)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)



minus 119906) (80)


where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 2 Continued


11986039

1 119906 = 119910 V =1 + 2+ (119905 minus 119909)

2

(1 + 1199052 + 1199092)22

119908 =

(1 + 1199052 + 1199092)32

3

119889V

119889119906= minus

2(1 + 2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2

(1 + 1199052 + 1199092)32

3

= 119891 (119906 V)

minus2(1 +

2+ (119905 minus 119909)

2)

4(1 + 1199052 + 1199092)

2minus

4(119905 + 119909)(1 + 2+ (119905 minus 119909)

2)

3(1 + 1199052 + 1199092)

3minus

2(119909119905 minus (1 + 1199052))

3(1 + 1199052 + 1199092)

2= 119892(119906 V)



119908 = sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

119889V

119889119906=

(cos2119909 + 2) sec3119909


2sec2119909

+((1 +



2sec2119909 sec2119909

( sec119909 + tan119909)3+

(cos119909 sin119909 + 22 tan119909)sec2119909

( sec119909 + tan119909)3

= 119891(119906 V)

(cos2119909 + 2) sec3119909


2sec2119909

+((1 +



2sec2119909= 119892(119906 V)


|119886| le


119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906=

119890minus119910

119886minus1

2+

+(119886 minus 1) 119890


2+

(6)






119905minus 119863119905119883 = minus(119863

119905120585)119863119905(see eg [17])


119889119906=

119863119905V

119863119905119906

= 119863V (7)


119863 = (119863119905119906)minus1

119863119905 (8)



1198862

36 |119886| le


1198862

36 |119886| le


119906 = 119890119910 V =

119886minus1 119908 =

119889V

119889119906= 119863V

119889119908

119889119906= 119863119908

1198892V

1198891199062= 1198632V

(9)


where119863 = 119890minus119910

(2+2)minus1

119863119905and119863119908 and119863


119890minus119910

119909 minus 119890minus119910

( + )

22(2+ )

= 119891 (119906 V 119908 119863V)

(119890minus2119910

119886minus3

(minus5+ (2119886 minus 5)


4

+2 (119886 minus 1) 2)) ((

2+ )3

)minus1

+ (119890minus2119910

119886minus3


3)

+2 119909 (2+)+


119910))

times ((2+ )3

)minus1

= 119892 (119906 V 119908 119863V)

(10)


1198862

36 |119886| le





1198862



3

31 1198603

35 1198604

31 and 119860

1

38in which two

of them 1198603

31and 119860

3


two 1198604

31and 119860

1


system


and1198603

35 we have


Consider 119860331


singular invariants

119906 = 119905 V = 119910 119908 = 119863V = = 0 (11)


= 119891 (119906 V 119863V) = 0 (12)


119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909 minus

119910 119863

3V =

119910

(13)


119909 =

119910

+

1

119891 (119906 V 119863V 119863

2V)

119910 = 119892 (119906 V 119863V 119863

2V)

(14)


3V where 119863 = 119863119905 For



119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909

119863

3V =

119910

(15)


31and

1198601


119906 = 119909 V = 119910 119863V =

1198632V =

2minus

3

(16)


minus1



=

+ 2119892 (119906 V 119863V) (17)


4


invariants

119908 =

11990912060110158401015840(119909)

4

minus3212060110158401015840(119909) +

2120601101584010158401015840

(119909)

5

1198633V =

119910

3+

(32minus

119909) minus 3

5

(18)

and for 119860138


119908 =

119909

3minus

32

24 119863

3V =

119910

3+

(32minus

119909) minus 3

5

(19)






4

31and

1198601






(a) 119906 V 119908119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus2

119908

119889119906119896minus2119889119896minus1V

119889119906119896minus1

(b) 119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus3

119908

119889119906119896minus3

119889119896V

119889119906119896

(20)


119896minus2119908119889119906119896minus2


119896minus3119908119889119906119896minus3

119889119896V119889119906119896


31 1198603

35 1198604

31 and 119860

1

38and (a) to

the remainder

3 Integrability



1198603

31⋍ 120597119909 119905120597119909 119910120597119909 = 119891 (119905 119910 ) = 0 (21)

Type II

1198603

35⋍ 120597119909 119905120597119909 119909120597119909 = 119892 (119905 119910 ) = 0 (22)

Type III

1198604

32⋍ 120597119909 119905120597119905+ 119909120597119909 119905120597119909

119905 = 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198603

34⋍ 120597119909 119905120597119909 minus120597119905+ 119909120597119909

119890119905 = 119891 (119910 )

= 119892 (119910 )

1198604

35⋍ 120597119909 119905120597119909 119909120597119909+ 120597119910

119890minus119910

= 119891 (119905 ) = 119892 (119905 )

1198601198863

36⋍ 120597119909 119905120597119909 (1 minus 119886) 119905120597

119905+ 119909120597119909

119905(2119886minus1)(119886minus1)

= 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198601198863

37⋍ 120597119909 119905120597119909 minus (1 + 119905

2) 120597119905+ (119886 minus 119905) 119909120597

119909

minus32

119890119886 tan 119905

= 119891 (119910 (1 + 1199052))

2119905 +(1 + 119905

2)

= 119892 (119910 (1 + 119905

2))

(23)










= 4 + 3119910 + 119910

3= 0 (24)






31or 11986035



32 1198603

34 1198604

35 119860119886336 and 119860

1198863







= 119890minus11990511991034 =

2119910minus1

minus 1199102 (25)


= 120597119909 1198832

=





119910 By


119905 = 119910 119909 = 119909 119910 = 119905 (26)


1199052

3




1

2sech21198881(119905 minus 119888

2)



119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

times tanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(27)



3





3

35












1 1198882 and 119888

3as well as 119888


constants

321 Uncoupled Cases

1198604

32 Let V = 120582(119906 119888


Table 2 we have

119905 = 120582 (119910 1198881) (28)


1198601

33 Let V = 120582(119906 119888



= 120582 (119910 1198881) (29)


=minus1

int 120582 (119910 1198881) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2

(30)


120582 (119910 1198881) int120582 (119910 119888

1) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2 = minus1 (31)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (32)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (33)


1

33

1198602

33 Let V = 120582(119906 119888



= plusmnradic2 + 22120582 (119910 119888

1) (34)


= minus int119891 (119910 120582 (119910 1198881)) 119889119910 + 119888

2 (35)


=2

(int119891(119910 120582(119910 1198881))119889119910 + 119888

2)2

minus 2120582 (119910 1198881)

(36)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (37)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (38)


2

33

1198603

34 Let V = 120582(119906 119888


= 120582 (119910 1198881) (39)

which implies that

119910 = 120601 (119905 1198881 1198882) (40)


119909 = intint 119890minus119905119891 (120601 120601

119905) 119889119905 119889119905 + 119888

3119905 + 1198884 (41)

which results in

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (42)


3

34

1198601

35 Let V = 120582(119906 119888


we obtain

= 120582 (119910 1198881) (43)


int119889119910

1198882exp int119891 (119910 120582 (119906 119888

1)) 119889119910

= 119905 + 1198883 (44)

Inverting we have

119910 = 120601 (119905 1198881 1198882 1198883) (45)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (46)


1

35

For119860435

1198601198863

36 and119860

1198863



34


322 Coupled Cases

1198601

31 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 ( 1198881) (47)


119909 = 120601 (119905 1198881 1198882 1198883) (48)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (49)


1

31

1198602


1198601

31

1198601

32 Let V = 120582(119906 119888



= 120582 (119890119910 1198881) (50)


= exp int ((120582 (119890119910 1198881) + 119890

119910119892 (119890119910 120582 (119890

119910 1198881)))

times119891 (119890119910 120582 (119890

119910 1198881)) 119889 (119890

119910))

times (119890119910(1+120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881))))minus1

(51)


119889 (119890119910)

119889119905

=2119890119910120582 (119890119910 1198881) (1 + 120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881)))

(120582 (119890119910 1198881) + 119890119910119892 (119890119910 120582 (119890119910 119888

1)))

(52)


119890119910= 120601 (119910 + 119888

2) (53)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (54)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (55)


1

32

1198602


35

1198603


1

33

1198603

33 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 (119910 minus

1

1198881) (56)


119891(119906 V) we arrive at

119910 = 120601(119910 minus1

1198881 1198882) (57)


=1

119910 minus 120601minus1 (119910 1198881 1198882) (58)


(119910 minus 120601minus1

(119910 1198881 1198882)) 120582 (120601

minus1(119910 1198881 1198882) 1198881) = 1 (59)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (60)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (61)


3

33

1198601


1

33

1198602


33

1198602

35 Let V = 120582(119906 119888



119890119910= 120582 ( 119888

1) (62)


= 120601 (119910 1198881 1198882) (63)


119910 = 120595 (119909 1198881 1198882 1198883) (64)


119909 = ℎ (119905 1198881 1198882 1198883 1198884) (65)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (66)


2

35


1198601198861


33

1198601198862


1198601

32

1198601198861


1198601


1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)


= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)


= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)


119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37




(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)




= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603



119906 = 119910 V = (78)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)



minus 119906) (80)


where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where119863 = 119890minus119910

(2+2)minus1

119863119905and119863119908 and119863


119890minus119910

119909 minus 119890minus119910

( + )

22(2+ )

= 119891 (119906 V 119908 119863V)

(119890minus2119910

119886minus3

(minus5+ (2119886 minus 5)


4

+2 (119886 minus 1) 2)) ((

2+ )3

)minus1

+ (119890minus2119910

119886minus3


3)

+2 119909 (2+)+


119910))

times ((2+ )3

)minus1

= 119892 (119906 V 119908 119863V)

(10)


1198862

36 |119886| le





1198862



3

31 1198603

35 1198604

31 and 119860

1

38in which two

of them 1198603

31and 119860

3


two 1198604

31and 119860

1


system


and1198603

35 we have


Consider 119860331


singular invariants

119906 = 119905 V = 119910 119908 = 119863V = = 0 (11)


= 119891 (119906 V 119863V) = 0 (12)


119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909 minus

119910 119863

3V =

119910

(13)


119909 =

119910

+

1

119891 (119906 V 119863V 119863

2V)

119910 = 119892 (119906 V 119863V 119863

2V)

(14)


3V where 119863 = 119863119905 For



119906 = 119905 V = 119910 119863V =

1198632V = 119908 =

119909

119863

3V =

119910

(15)


31and

1198601


119906 = 119909 V = 119910 119863V =

1198632V =

2minus

3

(16)


minus1



=

+ 2119892 (119906 V 119863V) (17)


4


invariants

119908 =

11990912060110158401015840(119909)

4

minus3212060110158401015840(119909) +

2120601101584010158401015840

(119909)

5

1198633V =

119910

3+

(32minus

119909) minus 3

5

(18)

and for 119860138


119908 =

119909

3minus

32

24 119863

3V =

119910

3+

(32minus

119909) minus 3

5

(19)






4

31and

1198601






(a) 119906 V 119908119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus2

119908

119889119906119896minus2119889119896minus1V

119889119906119896minus1

(b) 119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus3

119908

119889119906119896minus3

119889119896V

119889119906119896

(20)


119896minus2119908119889119906119896minus2


119896minus3119908119889119906119896minus3

119889119896V119889119906119896


31 1198603

35 1198604

31 and 119860

1

38and (a) to

the remainder

3 Integrability



1198603

31⋍ 120597119909 119905120597119909 119910120597119909 = 119891 (119905 119910 ) = 0 (21)

Type II

1198603

35⋍ 120597119909 119905120597119909 119909120597119909 = 119892 (119905 119910 ) = 0 (22)

Type III

1198604

32⋍ 120597119909 119905120597119905+ 119909120597119909 119905120597119909

119905 = 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198603

34⋍ 120597119909 119905120597119909 minus120597119905+ 119909120597119909

119890119905 = 119891 (119910 )

= 119892 (119910 )

1198604

35⋍ 120597119909 119905120597119909 119909120597119909+ 120597119910

119890minus119910

= 119891 (119905 ) = 119892 (119905 )

1198601198863

36⋍ 120597119909 119905120597119909 (1 minus 119886) 119905120597

119905+ 119909120597119909

119905(2119886minus1)(119886minus1)

= 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198601198863

37⋍ 120597119909 119905120597119909 minus (1 + 119905

2) 120597119905+ (119886 minus 119905) 119909120597

119909

minus32

119890119886 tan 119905

= 119891 (119910 (1 + 1199052))

2119905 +(1 + 119905

2)

= 119892 (119910 (1 + 119905

2))

(23)










= 4 + 3119910 + 119910

3= 0 (24)






31or 11986035



32 1198603

34 1198604

35 119860119886336 and 119860

1198863







= 119890minus11990511991034 =

2119910minus1

minus 1199102 (25)


= 120597119909 1198832

=





119910 By


119905 = 119910 119909 = 119909 119910 = 119905 (26)


1199052

3




1

2sech21198881(119905 minus 119888

2)



119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

times tanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(27)



3





3

35












1 1198882 and 119888

3as well as 119888


constants

321 Uncoupled Cases

1198604

32 Let V = 120582(119906 119888


Table 2 we have

119905 = 120582 (119910 1198881) (28)


1198601

33 Let V = 120582(119906 119888



= 120582 (119910 1198881) (29)


=minus1

int 120582 (119910 1198881) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2

(30)


120582 (119910 1198881) int120582 (119910 119888

1) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2 = minus1 (31)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (32)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (33)


1

33

1198602

33 Let V = 120582(119906 119888



= plusmnradic2 + 22120582 (119910 119888

1) (34)


= minus int119891 (119910 120582 (119910 1198881)) 119889119910 + 119888

2 (35)


=2

(int119891(119910 120582(119910 1198881))119889119910 + 119888

2)2

minus 2120582 (119910 1198881)

(36)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (37)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (38)


2

33

1198603

34 Let V = 120582(119906 119888


= 120582 (119910 1198881) (39)

which implies that

119910 = 120601 (119905 1198881 1198882) (40)


119909 = intint 119890minus119905119891 (120601 120601

119905) 119889119905 119889119905 + 119888

3119905 + 1198884 (41)

which results in

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (42)


3

34

1198601

35 Let V = 120582(119906 119888


we obtain

= 120582 (119910 1198881) (43)


int119889119910

1198882exp int119891 (119910 120582 (119906 119888

1)) 119889119910

= 119905 + 1198883 (44)

Inverting we have

119910 = 120601 (119905 1198881 1198882 1198883) (45)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (46)


1

35

For119860435

1198601198863

36 and119860

1198863



34


322 Coupled Cases

1198601

31 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 ( 1198881) (47)


119909 = 120601 (119905 1198881 1198882 1198883) (48)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (49)


1

31

1198602


1198601

31

1198601

32 Let V = 120582(119906 119888



= 120582 (119890119910 1198881) (50)


= exp int ((120582 (119890119910 1198881) + 119890

119910119892 (119890119910 120582 (119890

119910 1198881)))

times119891 (119890119910 120582 (119890

119910 1198881)) 119889 (119890

119910))

times (119890119910(1+120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881))))minus1

(51)


119889 (119890119910)

119889119905

=2119890119910120582 (119890119910 1198881) (1 + 120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881)))

(120582 (119890119910 1198881) + 119890119910119892 (119890119910 120582 (119890119910 119888

1)))

(52)


119890119910= 120601 (119910 + 119888

2) (53)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (54)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (55)


1

32

1198602


35

1198603


1

33

1198603

33 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 (119910 minus

1

1198881) (56)


119891(119906 V) we arrive at

119910 = 120601(119910 minus1

1198881 1198882) (57)


=1

119910 minus 120601minus1 (119910 1198881 1198882) (58)


(119910 minus 120601minus1

(119910 1198881 1198882)) 120582 (120601

minus1(119910 1198881 1198882) 1198881) = 1 (59)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (60)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (61)


3

33

1198601


1

33

1198602


33

1198602

35 Let V = 120582(119906 119888



119890119910= 120582 ( 119888

1) (62)


= 120601 (119910 1198881 1198882) (63)


119910 = 120595 (119909 1198881 1198882 1198883) (64)


119909 = ℎ (119905 1198881 1198882 1198883 1198884) (65)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (66)


2

35


1198601198861


33

1198601198862


1198601

32

1198601198861


1198601


1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)


= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)


= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)


119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37




(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)




= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603



119906 = 119910 V = (78)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)



minus 119906) (80)


where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












4

31and

1198601






(a) 119906 V 119908119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus2

119908

119889119906119896minus2119889119896minus1V

119889119906119896minus1

(b) 119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus3

119908

119889119906119896minus3

119889119896V

119889119906119896

(20)


119896minus2119908119889119906119896minus2


119896minus3119908119889119906119896minus3

119889119896V119889119906119896


31 1198603

35 1198604

31 and 119860

1

38and (a) to

the remainder

3 Integrability



1198603

31⋍ 120597119909 119905120597119909 119910120597119909 = 119891 (119905 119910 ) = 0 (21)

Type II

1198603

35⋍ 120597119909 119905120597119909 119909120597119909 = 119892 (119905 119910 ) = 0 (22)

Type III

1198604

32⋍ 120597119909 119905120597119905+ 119909120597119909 119905120597119909

119905 = 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198603

34⋍ 120597119909 119905120597119909 minus120597119905+ 119909120597119909

119890119905 = 119891 (119910 )

= 119892 (119910 )

1198604

35⋍ 120597119909 119905120597119909 119909120597119909+ 120597119910

119890minus119910

= 119891 (119905 ) = 119892 (119905 )

1198601198863

36⋍ 120597119909 119905120597119909 (1 minus 119886) 119905120597

119905+ 119909120597119909

119905(2119886minus1)(119886minus1)

= 119891 (119910 119905) 1 +119905

= 119892 (119910 119905)

1198601198863

37⋍ 120597119909 119905120597119909 minus (1 + 119905

2) 120597119905+ (119886 minus 119905) 119909120597

119909

minus32

119890119886 tan 119905

= 119891 (119910 (1 + 1199052))

2119905 +(1 + 119905

2)

= 119892 (119910 (1 + 119905

2))

(23)










= 4 + 3119910 + 119910

3= 0 (24)






31or 11986035



32 1198603

34 1198604

35 119860119886336 and 119860

1198863







= 119890minus11990511991034 =

2119910minus1

minus 1199102 (25)


= 120597119909 1198832

=





119910 By


119905 = 119910 119909 = 119909 119910 = 119905 (26)


1199052

3




1

2sech21198881(119905 minus 119888

2)



119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

times tanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(27)



3





3

35












1 1198882 and 119888

3as well as 119888


constants

321 Uncoupled Cases

1198604

32 Let V = 120582(119906 119888


Table 2 we have

119905 = 120582 (119910 1198881) (28)


1198601

33 Let V = 120582(119906 119888



= 120582 (119910 1198881) (29)


=minus1

int 120582 (119910 1198881) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2

(30)


120582 (119910 1198881) int120582 (119910 119888

1) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2 = minus1 (31)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (32)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (33)


1

33

1198602

33 Let V = 120582(119906 119888



= plusmnradic2 + 22120582 (119910 119888

1) (34)


= minus int119891 (119910 120582 (119910 1198881)) 119889119910 + 119888

2 (35)


=2

(int119891(119910 120582(119910 1198881))119889119910 + 119888

2)2

minus 2120582 (119910 1198881)

(36)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (37)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (38)


2

33

1198603

34 Let V = 120582(119906 119888


= 120582 (119910 1198881) (39)

which implies that

119910 = 120601 (119905 1198881 1198882) (40)


119909 = intint 119890minus119905119891 (120601 120601

119905) 119889119905 119889119905 + 119888

3119905 + 1198884 (41)

which results in

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (42)


3

34

1198601

35 Let V = 120582(119906 119888


we obtain

= 120582 (119910 1198881) (43)


int119889119910

1198882exp int119891 (119910 120582 (119906 119888

1)) 119889119910

= 119905 + 1198883 (44)

Inverting we have

119910 = 120601 (119905 1198881 1198882 1198883) (45)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (46)


1

35

For119860435

1198601198863

36 and119860

1198863



34


322 Coupled Cases

1198601

31 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 ( 1198881) (47)


119909 = 120601 (119905 1198881 1198882 1198883) (48)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (49)


1

31

1198602


1198601

31

1198601

32 Let V = 120582(119906 119888



= 120582 (119890119910 1198881) (50)


= exp int ((120582 (119890119910 1198881) + 119890

119910119892 (119890119910 120582 (119890

119910 1198881)))

times119891 (119890119910 120582 (119890

119910 1198881)) 119889 (119890

119910))

times (119890119910(1+120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881))))minus1

(51)


119889 (119890119910)

119889119905

=2119890119910120582 (119890119910 1198881) (1 + 120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881)))

(120582 (119890119910 1198881) + 119890119910119892 (119890119910 120582 (119890119910 119888

1)))

(52)


119890119910= 120601 (119910 + 119888

2) (53)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (54)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (55)


1

32

1198602


35

1198603


1

33

1198603

33 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 (119910 minus

1

1198881) (56)


119891(119906 V) we arrive at

119910 = 120601(119910 minus1

1198881 1198882) (57)


=1

119910 minus 120601minus1 (119910 1198881 1198882) (58)


(119910 minus 120601minus1

(119910 1198881 1198882)) 120582 (120601

minus1(119910 1198881 1198882) 1198881) = 1 (59)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (60)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (61)


3

33

1198601


1

33

1198602


33

1198602

35 Let V = 120582(119906 119888



119890119910= 120582 ( 119888

1) (62)


= 120601 (119910 1198881 1198882) (63)


119910 = 120595 (119909 1198881 1198882 1198883) (64)


119909 = ℎ (119905 1198881 1198882 1198883 1198884) (65)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (66)


2

35


1198601198861


33

1198601198862


1198601

32

1198601198861


1198601


1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)


= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)


= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)


119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37




(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)




= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603



119906 = 119910 V = (78)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)



minus 119906) (80)


where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= 4 + 3119910 + 119910

3= 0 (24)






31or 11986035



32 1198603

34 1198604

35 119860119886336 and 119860

1198863







= 119890minus11990511991034 =

2119910minus1

minus 1199102 (25)


= 120597119909 1198832

=





119910 By


119905 = 119910 119909 = 119909 119910 = 119905 (26)


1199052

3




1

2sech21198881(119905 minus 119888

2)



119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

times tanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(27)



3





3

35












1 1198882 and 119888

3as well as 119888


constants

321 Uncoupled Cases

1198604

32 Let V = 120582(119906 119888


Table 2 we have

119905 = 120582 (119910 1198881) (28)


1198601

33 Let V = 120582(119906 119888



= 120582 (119910 1198881) (29)


=minus1

int 120582 (119910 1198881) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2

(30)


120582 (119910 1198881) int120582 (119910 119888

1) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2 = minus1 (31)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (32)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (33)


1

33

1198602

33 Let V = 120582(119906 119888



= plusmnradic2 + 22120582 (119910 119888

1) (34)


= minus int119891 (119910 120582 (119910 1198881)) 119889119910 + 119888

2 (35)


=2

(int119891(119910 120582(119910 1198881))119889119910 + 119888

2)2

minus 2120582 (119910 1198881)

(36)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (37)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (38)


2

33

1198603

34 Let V = 120582(119906 119888


= 120582 (119910 1198881) (39)

which implies that

119910 = 120601 (119905 1198881 1198882) (40)


119909 = intint 119890minus119905119891 (120601 120601

119905) 119889119905 119889119905 + 119888

3119905 + 1198884 (41)

which results in

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (42)


3

34

1198601

35 Let V = 120582(119906 119888


we obtain

= 120582 (119910 1198881) (43)


int119889119910

1198882exp int119891 (119910 120582 (119906 119888

1)) 119889119910

= 119905 + 1198883 (44)

Inverting we have

119910 = 120601 (119905 1198881 1198882 1198883) (45)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (46)


1

35

For119860435

1198601198863

36 and119860

1198863



34


322 Coupled Cases

1198601

31 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 ( 1198881) (47)


119909 = 120601 (119905 1198881 1198882 1198883) (48)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (49)


1

31

1198602


1198601

31

1198601

32 Let V = 120582(119906 119888



= 120582 (119890119910 1198881) (50)


= exp int ((120582 (119890119910 1198881) + 119890

119910119892 (119890119910 120582 (119890

119910 1198881)))

times119891 (119890119910 120582 (119890

119910 1198881)) 119889 (119890

119910))

times (119890119910(1+120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881))))minus1

(51)


119889 (119890119910)

119889119905

=2119890119910120582 (119890119910 1198881) (1 + 120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881)))

(120582 (119890119910 1198881) + 119890119910119892 (119890119910 120582 (119890119910 119888

1)))

(52)


119890119910= 120601 (119910 + 119888

2) (53)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (54)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (55)


1

32

1198602


35

1198603


1

33

1198603

33 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 (119910 minus

1

1198881) (56)


119891(119906 V) we arrive at

119910 = 120601(119910 minus1

1198881 1198882) (57)


=1

119910 minus 120601minus1 (119910 1198881 1198882) (58)


(119910 minus 120601minus1

(119910 1198881 1198882)) 120582 (120601

minus1(119910 1198881 1198882) 1198881) = 1 (59)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (60)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (61)


3

33

1198601


1

33

1198602


33

1198602

35 Let V = 120582(119906 119888



119890119910= 120582 ( 119888

1) (62)


= 120601 (119910 1198881 1198882) (63)


119910 = 120595 (119909 1198881 1198882 1198883) (64)


119909 = ℎ (119905 1198881 1198882 1198883 1198884) (65)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (66)


2

35


1198601198861


33

1198601198862


1198601

32

1198601198861


1198601


1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)


= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)


= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)


119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37




(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)




= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603



119906 = 119910 V = (78)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)



minus 119906) (80)


where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1 1198882 and 119888

3as well as 119888


constants

321 Uncoupled Cases

1198604

32 Let V = 120582(119906 119888


Table 2 we have

119905 = 120582 (119910 1198881) (28)


1198601

33 Let V = 120582(119906 119888



= 120582 (119910 1198881) (29)


=minus1

int 120582 (119910 1198881) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2

(30)


120582 (119910 1198881) int120582 (119910 119888

1) 119891 (119910 120582 (119910 119888

1)) 119889119910 + 119888

2 = minus1 (31)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (32)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (33)


1

33

1198602

33 Let V = 120582(119906 119888



= plusmnradic2 + 22120582 (119910 119888

1) (34)


= minus int119891 (119910 120582 (119910 1198881)) 119889119910 + 119888

2 (35)


=2

(int119891(119910 120582(119910 1198881))119889119910 + 119888

2)2

minus 2120582 (119910 1198881)

(36)

which implies that

119910 = 120601 (119905 1198881 1198882 1198883) (37)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (38)


2

33

1198603

34 Let V = 120582(119906 119888


= 120582 (119910 1198881) (39)

which implies that

119910 = 120601 (119905 1198881 1198882) (40)


119909 = intint 119890minus119905119891 (120601 120601

119905) 119889119905 119889119905 + 119888

3119905 + 1198884 (41)

which results in

119909 = 120595 (119905 1198881 1198882 1198883 1198884) (42)


3

34

1198601

35 Let V = 120582(119906 119888


we obtain

= 120582 (119910 1198881) (43)


int119889119910

1198882exp int119891 (119910 120582 (119906 119888

1)) 119889119910

= 119905 + 1198883 (44)

Inverting we have

119910 = 120601 (119905 1198881 1198882 1198883) (45)


119909 = 120595 (119905 1198881 1198882 1198883 1198884) (46)


1

35

For119860435

1198601198863

36 and119860

1198863



34


322 Coupled Cases

1198601

31 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 ( 1198881) (47)


119909 = 120601 (119905 1198881 1198882 1198883) (48)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (49)


1

31

1198602


1198601

31

1198601

32 Let V = 120582(119906 119888



= 120582 (119890119910 1198881) (50)


= exp int ((120582 (119890119910 1198881) + 119890

119910119892 (119890119910 120582 (119890

119910 1198881)))

times119891 (119890119910 120582 (119890

119910 1198881)) 119889 (119890

119910))

times (119890119910(1+120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881))))minus1

(51)


119889 (119890119910)

119889119905

=2119890119910120582 (119890119910 1198881) (1 + 120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881)))

(120582 (119890119910 1198881) + 119890119910119892 (119890119910 120582 (119890119910 119888

1)))

(52)


119890119910= 120601 (119910 + 119888

2) (53)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (54)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (55)


1

32

1198602


35

1198603


1

33

1198603

33 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 (119910 minus

1

1198881) (56)


119891(119906 V) we arrive at

119910 = 120601(119910 minus1

1198881 1198882) (57)


=1

119910 minus 120601minus1 (119910 1198881 1198882) (58)


(119910 minus 120601minus1

(119910 1198881 1198882)) 120582 (120601

minus1(119910 1198881 1198882) 1198881) = 1 (59)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (60)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (61)


3

33

1198601


1

33

1198602


33

1198602

35 Let V = 120582(119906 119888



119890119910= 120582 ( 119888

1) (62)


= 120601 (119910 1198881 1198882) (63)


119910 = 120595 (119909 1198881 1198882 1198883) (64)


119909 = ℎ (119905 1198881 1198882 1198883 1198884) (65)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (66)


2

35


1198601198861


33

1198601198862


1198601

32

1198601198861


1198601


1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)


= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)


= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)


119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37




(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)




= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603



119906 = 119910 V = (78)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)



minus 119906) (80)


where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










322 Coupled Cases

1198601

31 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 ( 1198881) (47)


119909 = 120601 (119905 1198881 1198882 1198883) (48)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (49)


1

31

1198602


1198601

31

1198601

32 Let V = 120582(119906 119888



= 120582 (119890119910 1198881) (50)


= exp int ((120582 (119890119910 1198881) + 119890

119910119892 (119890119910 120582 (119890

119910 1198881)))

times119891 (119890119910 120582 (119890

119910 1198881)) 119889 (119890

119910))

times (119890119910(1+120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881))))minus1

(51)


119889 (119890119910)

119889119905

=2119890119910120582 (119890119910 1198881) (1 + 120582 (119890

119910 1198881) 119891 (119890

119910 120582 (119890

119910 1198881)))

(120582 (119890119910 1198881) + 119890119910119892 (119890119910 120582 (119890119910 119888

1)))

(52)


119890119910= 120601 (119910 + 119888

2) (53)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (54)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (55)


1

32

1198602


35

1198603


1

33

1198603

33 Let V = 120582(119906 119888


Table 2 we obtain

= 120582 (119910 minus

1

1198881) (56)


119891(119906 V) we arrive at

119910 = 120601(119910 minus1

1198881 1198882) (57)


=1

119910 minus 120601minus1 (119910 1198881 1198882) (58)


(119910 minus 120601minus1

(119910 1198881 1198882)) 120582 (120601

minus1(119910 1198881 1198882) 1198881) = 1 (59)

which implies that

119910 = ℎ (119905 1198881 1198882 1198883) (60)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (61)


3

33

1198601


1

33

1198602


33

1198602

35 Let V = 120582(119906 119888



119890119910= 120582 ( 119888

1) (62)


= 120601 (119910 1198881 1198882) (63)


119910 = 120595 (119909 1198881 1198882 1198883) (64)


119909 = ℎ (119905 1198881 1198882 1198883 1198884) (65)


119910 = 120595 (119905 1198881 1198882 1198883 1198884) (66)


2

35


1198601198861


33

1198601198862


1198601

32

1198601198861


1198601


1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)


= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)


= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)


119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37




(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)




= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603



119906 = 119910 V = (78)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)



minus 119906) (80)


where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198601198861


33

1198601198862


1198601

32

1198601198861


1198601


1198862

37

1198601198862

37 Let V = 120582(119906 119888


Table 2 we obtain


radic1 + 2

= 120582 (119910 + arctan 1198881) (67)


= tanint119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882 (68)


= 120582 (119906 1198881) sec 119886int

119891 (119906 120582 (119906 1198881)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198882

times exp119886int119891 (119906 120582 (119906 119888

1)) 119889119906

120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1))

+ 1198861198882

(69)

We have

119889119906

119889119905=

(120582 (119906 1198881) + 119891 (119906 120582 (119906 119888

1)))

120582 (119906 1198881)

(70)

In (68)ndash(70)

119906 = 119910 + arctan (71)

Also we find that

119906 = 120601minus1

(119905 + 1198883)

119910 + arctan = 120601minus1

(119905 + 1198883)

(72)


119906 = 120595 (119905 1198881 1198882 1198883) (73)

which shows that

119910 = ℎ (119905 1198881 1198882 1198883) (74)


119909 = 119896 (119905 1198881 1198882 1198883 1198884) (75)


1198862

37




(a) 119889119896minus1V

119889119906119896minus1=119891(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

119889119896minus2

119908

119889119906119896minus2= 119892(119906 V 119908

119889V

119889119906119889119908

1198891199061198892V

1198891199062

119889119896minus3

119908

119889119906119896minus3119889119896minus2V

119889119906119896minus2)

(b) 119889119896V

119889119906119896

=119891(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

d1199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

119889119896minus3

119908

119889119906119896minus3

=119892(119906 V119889V

1198891199061198892V

1198891199062 119908

1198893V

1198891199063119889119908

1198891199061198894V

1198891199064

119889119896minus4

119908

119889119906119896minus4119889119896minus1V

119889119906119896minus1)

(76)




= 119890minus11990511991034 =

2119910minus1

minus 1199102 (77)


= 120597119909 1198832

=


which is 1198603



119906 = 119910 V = (78)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (79)



minus 119906) (80)


where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119888lowast



119910 = 21198881

2sech21198881(119905 minus 1198882) (81)




119909 = 2048 1198881

18intint 119890

minus119905sech14 (1198881(119905 minus 1198882))

timestanh4 (1198881(119905 minus 1198882)) 119889119905 119889119905 + 119888

3119905 + 1198884

(82)


coordinates is

119903 + 120583119903

1199033= 0 (83)


119903 minus 1199032

+ 120583119903minus2

= 0 119903 + 2 119903 = 0 (84)


1198831= 120597119905 119883

2= 120597120579 119883

3= 119905120597119905minus

2

3119903120597119903 (85)


119905 = 119905 119909 = 120579 119910 =3

2ln 119903 (86)


+2

32

minus3

22

+3

2120583119890minus2

= 0 +4

3 = 0 (87)


1198831= 120597119905 119883

2= 120597119909 119883

3= 119905120597119905+ 120597119910 (88)

which is 1198601



119906 = 119890119910 V =

(89)


119889V

119889119906= 119892 (119906 V) 119908 = 119891 (119906 V) (90)


119889V

119889119906=

minus4V + 9 (120583V1199062) minus 9V3

2119906 minus 9 (120583119906) + 9V2119906 119908 = minus

4

3V (91)


1199064V2minus

9

21205831199062V2+

9

41199064V4= ℎ (92)


from (91) give

119889 (119890(23)119910

)


4ℎ

91198962+ 2120583119890(23)119910 minus 119890(43)119910

(119890(23)119910

)2

= 119896

(93)


119889119903


4ℎ

91198962+

2120583

119903minus

1

1199032 (94)

1199032 = 119896 (95)


119889120579 =119889119904


(96)





119903 minus 1199032

=1

1199033119891 (120579 119903

2)

119903 + 2 119903 =1

1199033119892 (120579 119903

2)

(97)


1198831= 120597119905 119883

2= 2119905120597119905+ 119903120597119903 119883

3= 1199052120597119905+ 119905119903120597119903 (98)


38in our



2 we find that


119872 =1

2(1199032)2

minus intℎ (120579) 119889120579 (99)



4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4 Conclusion



31and119860

3

35






Remarks


119886119887119899

119894119895 (the 119895th algebra of


1


119905=

120597120597119905 120597119909

= 120597120597119909 120597119910


119894 where 119894 is



4

31and

1198601



4

31≃

120597119905 119909120597119905 120601(119909)120597



1198892V1198891199062 = minus




31and 119860

3

35 due to the extra





31and 119860

1



3

31and 119860

3




31and 119860

1



Acknowledgments




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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