Research ArticleApproximate Symmetry Analysis and Approximate ConservationLaws of Perturbed KdV Equation

Yu-Shan Bai and Qi Zhang

Department of Mathematics Inner Mongolia University of Technology Hohhot 010051 China

Correspondence should be addressed to Yu-Shan Bai mbaiyushanimuteducn

Received 9 February 2018 Revised 11 August 2018 Accepted 27 August 2018 Published 9 September 2018

Academic Editor Boris G Konopelchenko

Copyright copy 2018 Yu-Shan Bai and Qi Zhang 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

Approximate symmetries which are admitted by the perturbedKdV equation are obtainedThe optimal systemof one-dimensionalsubalgebra of symmetry algebra is obtained The approximate invariants of the presented approximate symmetries and some newapproximately invariant solutions to the equation are constructedMoreover the conservation laws have been constructed by usingpartial Lagrangian method

1 Introduction

The partial differential equations (PDEs) with small param-eter have been arisen in mathematics physics mechanicsetc For these perturbed equations the finding of analyticsolutions conservation laws symmetries etc is essential inthese related fields Traditionally various perturbation meth-ods are used to solve such equations In addition emerginginnovative methods such as the inverse scattering transfor-mation method homotopy perturbation method Adomiandecomposition method and approximate symmetry methodhave been developed fast in the past few decades [1ndash4]The approximate symmetry method which is developed byBaikov Gazizov and Ibragimov [3 4] in the 1980s shows aneffectiveness to obtain approximate solutions to a perturbedPDEs The idea behind this development was the extensionof Liersquos theory in which a small perturbation of the originalequation is encountered The new method maintains theessential features of the standard Lie symmetry method andprovides us with the most widely applicable technique tofind the approximate solutions to a perturbed differentialequations [5 6]

The construction of conservation laws is one of the keyapplications of symmetries to physical problems [7 8] Butfor perturbed systems one hindrance is how to constructconservation laws Combined with the Noether theory andLagrangian principle the approximate symmetry method

yields the approximate conservation laws for a perturbedPDEs There are mainly two methods One approach basedon generalizations of the Noetherrsquos theorem to perturbedequations is given to get approximate conservation lawsvia the approximate Noether symmetries associated with aLagrangian of the perturbed equations [5 9] This methoddepends on the existence of the Lagrangian functional forunderlying differential equations The other approach (alsocalled partial Lagrangian method) which is presented byJohnpillai Kara and Mahomed [10 11] demonstrates howone can construct approximate conservation laws of Eular-Type PDEs via approximate Noether-type symmetry opera-tors associated with partial Lagrangian Partial Lagrangianmethod can construct approximate conservation laws ofperturbed equations via approximate operators that are notnecessarily approximate symmetry operators of the under-lying system of equations This method is used for moregeneral equations as even the equations donot admit essentialLagrangian

In this paper we consider perturbed KdV equation

119906119905 minus 6119906119906119909 + 119906119909119909119909 + 120576120572 (119906 + 2119909119906119909) minus 120576120573 (2119906 + 119909119906119909)= 0 (1)

where 120576 is a small parameter while 120572 and 120573 are arbitraryconstantsObviously the equation has noLagrangian becauseof its odd order In [12] E S Benilov and B A Malomed

HindawiAdvances in Mathematical PhysicsVolume 2018 Article ID 4743567 11 pageshttpsdoiorg10115520184743567

2 Advances in Mathematical Physics

discussed the equation based on the inverse scattering trans-formation and showed its integrability When 120573 = 2120572 (1)has important applications in the description of nonlinear ionacoustic waves in an inhomogeneous plasma For many otherapplications of (1) please refer to [2 12 13] and the referencestherein

In this article with the application of approximate sym-metry method and partial Lagrangian method we will inves-tigate (1) and show its all first-order approximate symmetriesFurthermore we will also construct several approximatesolutions and approximate conservation laws of the equationThe rest of the paper is organized as follows Section 2gives some basic concepts and notations Section 3 analyzesapproximate symmetries of (1) by applying the approx-imate symmetry method developed by Baikov Gazizovand Ibragimov We compute the optimal system of thepresented approximate symmetries In Section 4 we generatethe approximate invariants of presented approximate symme-tries and construct corresponding approximately invariantsolutions Section 5 the final part presents approximateconservation laws via partial Lagrangian method

2 Notations and Definitions

We will use the following notations and definitions Let 119866 bea one-parameter approximate transformation group

119894 asymp 119891 (119909 119886 120576) equiv 1198911198940 (119911 119886) + 1205761198911198941 (119911 119886) 119894 = 1 119873 (2)

An approximate equation

119865 (119911 120576) equiv 1198650 (119911) + 1205761198651 (119911) asymp 0 (3)

is said to be approximately invariant with respect to 119866 oradmits 119866 if

119865 ( 120576) asymp 119865 (119891 (119911 119886 120576) 120576) = 119900 (120576) (4)

whenever 119911 = (1199111 119911119873) satisfies (3)If 119911 = (119909 119906 119906(1) 119906(119896)) where independent variables119909 = (1199091 119909119899) dependent variables 119906 and 119906(119896) denote

the collections of all 119896th-order partial derivatives then (3)becomes an approximate differential equation of 119896 and 119866 isan approximate symmetry group of the differential equation

Theorem 1 (see [5]) Equation (3) is approximately invariantunder the approximate transformation group (2) with thegenerator

119883 = 1198830 + 1205761198831 equiv 1205851198940 (119911) 120597120597119911119894 + 1205761205851198941 (119911) 120597

120597119911119894 (5)

if and only if

[119883(119896)119865 (119911 120576)]119865asymp0

= 119900 (120576) (6)

or

[119883(119896)0 1198650 (119911) + 120576 (119883(119896)1 1198650 (119911) + 119883(119896)0 1198651 (119911))](3) = 119900 (120576) (7)

in which 119896 is order of equation and 119883(119896) is 119896 minus 119905ℎ orderprolongation of 119883 The operator (5) satisfying (7) is called aninfinitesimal approximate symmetry or an approximate oper-ator admitted by (3) Accordingly (7) is termed the determin-ing equation for approximate symmetries

Remark 2 The determining equation (7) can be written asfollows

119883(119896)0 1198650 (119911) = 120582 (119911) 1198650 (119911) (8)

119883(119896)1 1198650 (119911) + 119883(119896)0 1198651 (119911)) = 120582 (119911) 1198651 (119911) (9)

The factor 120582(119911) is determined by (8) and then substituted in(9) The latter equation must hold for all solutions of 1198650(119911) =0 Comparing (8) with the determining equation of exactsymmetries we obtain the following statement

Theorem 3 (see [5]) If (3) admits an approximate transfor-mation group with the generator 119883 = 1198830 +1205761198831 where1198830 = 0then the operator

1198830 = 1205851198940 120597120597119911119894 (10)

is an exact symmetry of

1198650 (119911) = 0 (11)

Remark 4 It is manifested from (8) and (9) that if 1198830 is anexact symmetry of (11) then 119883 = 1205761198830 is an approximatesymmetry of (3)

Definition 5 (see [5]) Equations (11) and (3) are termed anunperturbed equation and a perturbed equation respectivelyUnder the conditions of Theorem 3 the operator1198830 is calleda stable symmetry of the unperturbed equation (11) The cor-responding approximate symmetry generator 119883 = 1198830 + 1205761198831for the perturbed equation (10) is called a deformation of theinfinitesimal symmetry 1198830 of (11) caused by the perturbation1205761198651(119909) In particular if themost general symmetry Lie algebraof Eq(11) is stable we say that the perturbed equation (3)inherits the symmetries of the unperturbed equation

3 Approximate Symmetry Analysis

31 Exact Symmetries Let us write the approximate groupgenerator in the form

119883 = 1198830 + 1205761198831= (1205850 + 1205761205851) 120597

120597119909 + (1205910 + 1205761205911) 120597120597119910 + (1205780 + 1205761205781) 120597

120597119906 (12)

where 120585119894 120591119894 and 120578119894 (119894 = 0 1) are unknown functions of 119909 119905and 119906

Solving the determining equation

119883(119896)0 1198650 (119911)100381610038161003816100381610038161198650(119911)=0 = 0 (13)

Advances in Mathematical Physics 3

for the exact symmetries of the unperturbed equation 119906119905 minus6119906119906119909 + 119906119909119909119909 = 0 we can get

1205780119905 = 01205780119909 = 01205910119909 = 01205910119906 = 0

1205780119906 + 21205850119909 = 01205850119909119909 = 01205850119906 = 0

61205780 + 1205850119905 + 121199061205850119909 = 031205850119909 minus 1205910119905 = 0

(14)

Then we obtain 1205850 = 1198881119909 minus 61198882119905 + 1198883 1205910 = 31198881119905 + 1198882 1205780 =minus21198881119906 + 1198882 where 1198881 1198882 1198883 1198884 are arbitrary constants Hence1198830 = (1198881119909 minus 61198882119905 + 1198883) 120597

120597119909 + (31198881119905 + 1198884) 120597120597119905

+ (minus21198881119906 + 1198882) 120597120597119906

(15)

In other words 119906119905 minus 6119906119906119909 + 119906119909119909119909 = 0 admits the four-dimen-sional Lie algebra with the basis

11988310 = 119909 120597120597119909 + 3119905 120597120597119905 minus 2119906 120597

120597119906 11988320 = minus6119905 120597

120597119909 + 120597120597119906

11988330 = 120597120597119909

11988340 = 120597120597119906

(16)

32 Approximate Symmetries and Optimal System

321 Approximate Symmetries The Cases 120572 and 120573 Are Arbi-trary First we need to determine the auxiliary function 119867by virtue of (15) by

119867 = 1120576 [119883(119896)0 (1198650 (119911) + 1205761198651 (119911))100381610038161003816100381610038161198650(119911)+1205761198651(119911)=0] (17)

Substituting the expression (15) of the generator 1198830 intoabove equation we obtain the auxiliary function

119867 = 1198882120572 minus 21198882120573 + (31198881120572 minus 61198881120573) 119906+ (21198883120572 minus 121198882120572119905 + 61198881120572119909 minus 1198883120573 + 61198882120573119905 minus 31198881120573119909)sdot 119906119909

(18)

Now calculate the operators 1198831 by solving the inhomoge-neous determining equation for deformations

119883(119896)1 1198650 (119911)100381610038161003816100381610038161198650(119911)=0 + 119867 = 0 (19)

Above determining equation yields

1198882120572 minus 21198882120573 minus 61198881120572119906 minus 61198881120573119906 + 1205781119905 = 021198881120572 minus 21205781119909 = 01205911119909 = 01205781119906 = 01205781119906 + 21205851119909 = 01205851119909119909 = 01205911119906 = 031205851119909 minus 1205911119905 = 021198883120572 minus 1198883120573 + 61198881120572119909 minus 31198881120573119909 minus 121198882120572119905 + 61198882120573119905 minus 61205781

minus 1205851119905 minus 121199061205851119909 = 0

(20)

Solving this system we obtain

1198831 = [minus3 (1198881120572 + 1198881120573) 119905119909 + (21198883120572 minus 1198883120573 minus 61198891) 119905minus 3 (1198882120572 + 1198882120573) 1199052 + 1198892] 120597

120597119909+ [minus9 (1198881120572 + 1198881120573) 11990522 + 1198893] 120597

120597119905 + [311988811205721199092+ (21198882120573 + 61198881120572119906 minus 1198882120572 + 61198881120573119906) 119905 + 1198891] 120597

120597119906

(21)

Then we obtain the following approximate symmetries of (1)

119883 = [1198881119909 minus 61198882119905 + 1198883 + 120576 [minus3 (1198881120572 + 1198881120573) 119905119909+ (21198883120572 minus 1198883120573 minus 61198891) 119905 minus 3 (1198882120572 + 1198882120573) 1199052

+ 1198892]] 120597120597119909 + [31198881119905 + 1198882 + 120576 [minus9 (1198881120572 + 1198881120573) 11990522

+ 1198893]] 120597120597119905 + [minus21198881119906 + 1198882 + 120576 [311988811205721199092

+ (21198882120573 + 61198881120572119906 minus 1198882120572 + 61198881120573119906) 119905 + 1198891]] 120597120597119906

(22)

and we have

V1 = 119909 120597120597119909 + 3119905 120597120597119905 minus 2119906 120597

120597119906 + 120576 [minus3 (120572 + 120573) 119905119909 120597120597119909

minus 92 (120572 + 120573) 1199052 120597120597119905 + (31205721199092 + 6120572119906119905 + 6120573119906119905) 120597

120597119906]

V2 = minus6119905 120597120597119909 + 120597

120597119906 + 120576 [minus3 (120572 + 120573) 1199052 120597120597119909+ (2120573 minus 120572) 119905 120597

120597119906]

V3 = 120597120597119909 + 120576 [(2120572 minus 120573) 119905 120597

120597119909]

4 Advances in Mathematical Physics

V4 = 120597120597119906

V5 = 120576V2V6 = 120576V3V7 = 120576V4

(23)

Tables 1 2 and 3 of commutator evaluated in the first-order of precision show that above operators span a seven-dimensional approximate Lie algebra 1198717322 Approximate Symmetries The Case 120572 = 2120573 When 120572 =2120573 the equation admits seven-dimensional approximate Liealgebra 1198717 as follows

V1 = 119909 120597120597119909 + 3119905 120597120597119905 minus 2119906 120597

120597119906 + 120576 [minus9120572119905119909 120597120597119909 minus 27

2 1205721199052 120597120597119905+ (31205721199092 + 6120572119906119905 + 6120573119906119905) 120597

120597119906]

V2 = minus6119905 120597120597119909 + 120597120597119906 + 120576 [minus91205721199052 120597120597119909]

V3 = 120597120597119909 + 120576 [3120572119905 120597

120597119909] V4 = 120597

120597119906V5 = 120576V2V6 = 120576V3V7 = 120576V4

(24)

323 Approximate Symmetries and Optimal System The Case120572 = minus120573 When 120572 = minus120573 the equation admits seven-dimen-sional approximate Lie algebra 1198717 as follows

V1 = 119909 120597120597119909 + 3119905 120597120597119905 minus 2119906 120597

120597119906 + 120576 [31205721199092120597120597119906]

V2 = minus6119905 120597120597119909 + 120597120597119906 + 120576 [minus3120572119905 120597120597119906]

V3 = 120597120597119909 + 120576 [3120572119905 120597

120597119909] V4 = 120597

120597119906V5 = 120576V2V6 = 120576V3V7 = 120576V4

(25)

It is worth nothing that the seven-dimensional approxi-mate Lie algebra 1198717 = 119892 is solvable and its finite sequence ofideals is as follows

0 sub ⟨V6⟩ sub ⟨V5 V6⟩ sub ⟨V3 V5 V6⟩ sub ⟨V2 V3 V5 V6⟩sub ⟨V2 V3 V4 V5 V6⟩ sub ⟨V2 V3 V4 V5 V6 V7⟩ sub 119892 (26)

In the following we will construct the optimal system ofabove Lie algebra 1198717Themethod used here for obtaining theone-dimensional optimal system of subalgebras is that givenin [7] This approach is taking a general element from theLie algebra and reducing it to its simplest equivalent formby applying carefully chosen adjoint transformations that aredefined as follows

Definition 6 (see [7]) Let119866 be a Lie groupAnoptimal systemof s-parameter subgroups is a list of conjugacy inequivalents-parameter subalgebras with the property that any othersubgroup is conjugate to precisely one subgroup in the listSimilarly a list of s-parameter subalgebras forms an optimalsystem if every s-parameter subalgebra of 119892 is equivalent to aunique member of the list under some element of the adjointrepresentation ℎ = 119860119889(119892(ℎ)) 119892 isin 119866Theorem7 (see [7]) Let119867 and be connected s-dimensionalLie subgroups of the Lie group 119866 with corresponding Liesubalgebras ℎ and ℎ of the Lie algebra 119892 of119866Then119867 = 119892119867119892minus1are conjugate subgroups if and only if ℎ = 119860119889(119903(ℎ)) are con-jugate subalgebras

By Theorem 7 the problem of finding an optimal systemof subgroups is equivalent to that of finding an optimal systemof subalgebras To compute the adjoint representation we usethe Lie series

119860119889 (exp (120583V119894)) V119895 = V119895 minus 120583 [V119894 V119895] + 12058322 [V119894 [V119894 V119895]]

minus sdot sdot sdot = exp (119886119889 (minus120583V119894)) V119895(27)

where [V119894 V119895] is the commutator for the Lie algebra and 120583 isa parameterand 119894 119895 = 1 7 In this manner we constructTable 4 with the (119894 119895)th entry indicating 119860119889(exp(120583V119894))V119895Theorem 8 An optimal system of one-dimensional approx-imate symmetry algebra (case 120572 = minus120573) of equation (1) isprovided by V7 V6 V1 + 119887V6 V3 + 119887V7 V2 + 119887V5 + 119888V7 V5 + 119887V3 +119888V7 V4 + 119887V2 + 119888V5 + 119889V7Proof Considering the approximate symmetry algebra g of(1) whose adjoint representation was determined in theTable 4 our task is to simplify as many of the coefficients 119886119894 aspossible through judicious applications of adjoint maps to119881119894so that 119881119894 is equivalent 1198811015840119894 under the adjoint representation

Given a nonzero vector

1198811 = 1198861V1 + 1198862V2 + 1198863V3 + 1198864V4 + 1198865V5 + 1198866V6 + 1198867V7 (28)

Advances in Mathematical Physics 5

Table 1 Approximate commutators of approximate symmetry algebra of (1) (119886 = 2120572 minus 120573 119887 = 2120573 minus 120572 119888 = 120572 + 120573 119889 = 31205722)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus119889V5 minus V3 minus3V4 + 3120576119888V1 2V5 minusV6 minus3V7V2 minus2V2 0 0 6V3 minus 119887V5 0 0 6V6V3 119889V5 + V3 0 0 minus119886V6 0 0 0V4 3V4 minus 3120576119888V1 minus6V3 + 119887V5 119886V6 0 minus6V6 0 0V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 2 Approximate commutators of approximate symmetry algebra of (1) (120572 = 2120573 119886 = 3120572)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus1198862 V5 minus V3 minus3V4 + 3120576119886V1 2V5 minusV6 minus3V7

V2 minus2V2 0 0 minus6V3 + 52 V5 0 0 6V6

V3 V3 + 52V5 0 0 minus119886V6 0 0 0

V4 3V4 minus 3120576119886V1 6V3 minus 52 V5 119886V6 0 minus6V6 0 0

V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 3 Approximate commutators of approximate symmetry algebra of (1) ( 120572 = minus120573 119886 = 3120572)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus1198862 V5 minus V3 minus3V4 2V5 minusV6 minus3V7

V2 minus2V2 0 0 6V3 + 119886V5 0 0 6V6V3

1198862 V5 + V3 0 0 minus119886V6 0 0 0

V4 3V4 minus6V3 minus 119886V5 119886V6 0 minus6V6 0 0V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 4 Adjoint representation of approximate symmetry of (1)

119860119889 V1 V2 V3 V4 V5 V6 V7V1 V1 119890minus2120583V2 119890minus120583V3 + 119886

2 119890minus120583V5 minus1198862 V5 1198903120583V4 119890minus2120583V5 119890minus120583V6 1198903120583V7

V2 V1 + 2120583V2 V2 V3 V4 minus 120583(6V3 + 119886V5) V5 V6 V7 minus 6120583V6V3 V1 minus 120583(119886V52 + V3) V2 V3 V4 + 120583(119886V6) V5 V6 V7V4 V1 minus 3120583V4 V2 + 120583(6V3 + 119886V5) V3 minus 119886120583V6 V4 V5 + 6120583V6 V6 V7V5 V1 + 2120583V5 V2 V3 V4 minus 6120583V6 V5 V6 V7V6 V1 minus 120583V6 V2 V3 V4 V5 V6 V7V7 V1 minus 3120583V7 V2 + 6120583V7 V3 V4 V5 V6 V7

First suppose that 1198861 = 0 Scaling 1198811 if necessary we canassume that 1198861 = 1 As for the 7th column of the Table 4we have

11988110158401 = 119860119889(exp ((11988643 V4))) ∘ 119860119889(exp ((minus11988622 V2)))∘ 119860119889 (exp ((11988673 V7)))

∘ 119860119889 (exp ((1198863 + 211988621198864) V3)))∘ 119860119889 (exp ((minus11988652 + 11988611988634 + 1198861198862119886412 ) V5)))1198811

= V1 + (1198866 minus 119886119886311988643 + 211988641198865 + 311988621198867) V6(29)

6 Advances in Mathematical Physics

The remaining approximate one-dimensional subalgebras arespanned by vectors of the above form with 1198861 = 0 If 1198864 = 0we have

1198812 = 1198862V2 + 1198863V3 + V4 + 1198865V5 + 1198866V6 + 1198867V7 (30)

Next we act on 1198812 to cancel the coefficients of V3 and V6 asfollows

11988110158402 = 119860119889(exp((minus119886361198862 V4)))

∘ 119860119889 (exp((minus119886561198862 +11988611988623361198862 +

11988666 ) V5)))1198812= 1198862V2 + V4 + (1198865 minus 11988611988636 V5 + 1198867V7

(31)

If 1198861 1198864 = 0 and 1198862 = 0 the nonzero vector1198813 = V2 + 1198863V3 + 1198865V5 + 1198866V6 + 1198867V7 (32)

is equivalent to

11988110158403 = 119860119889(exp ((minus11988636 V4)))

∘ 119860119889 (exp((minus11988666 + minus1198861198862336 + 119886611988656 ) V7)))1198813= V2 + (1198865 minus 11988611988636 ) V5 + 1198867V7

(33)

If 1198861 1198862 1198864 = 0 and 1198865 = 0 we scale to make 1198865 = 1 and then

1198814 = 1198863V3 + V5 + 1198866V6 + 1198867V7 (34)

is equivalent to 11988110158404 under the adjoint representation11988110158404 = 119860119889(exp(( 119886661198867 V2)))1198814 = 1198863V3 + V5 + 1198867V7 (35)

If 1198861 1198862 1198864 1198865 = 0 and 1198863 = 0 in the same way as before thenonzero vector

1198815 = V3 + 1198866V6 + 1198867V7 (36)

can be simplified as

11988110158405 = 119860119889(exp(( 119886661198867 V2)))1198812 = V3 + 1198867V7 (37)

If 1198861 1198862 1198863 1198864 1198865 = 0 and 1198867 = 0 we can act

1198816 = 1198866V6 + V7 (38)

by119860119889(exp(((11988666)V2))) to cancel the coefficient of V6 leadingto

11988110158406 = 119860119889 (exp ((11988666 V2)))1198816 = V7 (39)

The last remaining case occurs when 1198861 1198862 1198863 1198864 1198865 1198867 =0 and 1198866 = 0 for which our earlier simplifications wereunnecessary because the only remaining vectors are themultiples of 1198866 on which the adjoint representation actstrivially

4 Approximately Invariant Solutions

In this section we use two different techniques to constructnew approximate solutions of (1) when 120572 = minus12057341 Approximately Invariant Solutions I In the beginning ofthis section we compute an approximately invariant solutionbased on the 119883 = V2 The approximate invariants for 119883 aredetermined by

119883 (119869) = (minus6119905 120597120597119909 + 120597

120597119906 + 120576 (minus3120572119905 120597120597119906)) (1198690 + 1205761198691)= 119900 (120576)

(40)

Equivalently

(minus6119905 120597120597119909 + 120597120597119906) (1198690) = 0

(minus3120572119905 120597120597119906) (1198691) + (minus6119905 120597120597119909 + 120597120597119906) (1198690) = 0

(41)

Thefirst equation has two functionally independent solutions1198690 = 119905 and 1198690 = 1199096119905 + 119906 The simplest solutions of thesecond equation are respectively 1198691 = 0 and 1198691 = minus1205721199092 + 119905Therefore we have two independent invariants to 1199096119905 + 119906 +120576(minus1205721199092 + 119905) and 120593(119905) with respect to119883

Letting 1199096119905 + 119906 + 120576(3120572119905119906) = 120593(119905) we obtain 119906 = 120593(119905) minus1199096119905minus120576(minus1205721199092+ 119905) for the approximately invariant solutionsTherefore we can obtain

119906 (119909 119905) asymp minus 1199096119905 + Γ0 (119905) + 120576 (minus119905 minus 120572119909

2 + Γ1 (119905)+ Γ0 (119905) Γ10158401 (119905)

(42)

Putting (42) into (1) we obtain

Γ0 (119905) = 1198621119879Γ1 (119905) = 11986221199053 + 119905 + 1198622119905

(43)

where 1198621 and 1198622 are arbitrary constants Hence we obtainedthe approximately invariant solution to (1)

119906 (119909 119905) asymp minus 1199096119905 +

1198621119905 + 120576 (1198622119905 minus 1205721199092 ) (44)

In this manner we compute approximate invariants withrespect to the generators of Lie algebra and optimal systemas shown in Table 5

42 Approximately Invariant Solution II Now we apply adifferent technique to find approximately invariant solutionsfor (1) We will begin with one exact solution

119906 = 119908 (120577) = minus12

1cosh2120577 (45)

Advances in Mathematical Physics 7

Table 5 Approximate invariants with respect to the generators of Lie algebra and optimal system

Operator Approximate Invariants

V11199051199093 1199092119906 minus 120576(1205721199093

2 )V2 119905 1199096119905 + 119906 + 120576 (minus1205721199092 + 119905)V3 119905 119906V4 119909 119906V5 119905 1199096119905 + 119906V6 119905 119906V7 119909 119906V1 + 119887V6 119905

1199093 + 120576(119887119909minus4119905) 1199092119906 minus 120576(12057211990932 + 2119887119909

3119906 )V3 + 119887V7 119905 minus 120576(119887119909) 119906V4 + 119887V2 + 119888V5 + 119889V7 1199052

2 + 1199096119887 + 120576 (( 119888119887 minus 119889) 119905) 119887119906 minus 119905 + 120576(31205721199052

2 + 119889119905)V2 + 119887V5 + 119888V7 119905 minus 120576 (119906119888 )

1199096119905 + 119906 + 120576(3120572119905119906 minus 1198881199092

721199053 )V5 + 119887V3 + 119888V7 119905 minus 120576 (119888119909119887 ) 119906

of the unperturbed equation 119906119905 minus 6119906119906119909 + 119906119909119909119909 = 0 Here 120577 =(119909 minus 119905)2 is an invariant of the group with the generator

1198830 = 120597120597119905 +

120597120597119909 (46)

The function 119908 given by (45) is invariant under the operator(46)

Using the generators 1198831 = 120597120597119909 admitted by 119906119905 minus 6119906119906119909 +119906119909119909119909 = 0 we will take the approximate symmetry

119883 = 1198830 + 1205761198831 = 120597120597119905 +

120597120597119909 + 120576 120597

120597119909 (47)

and use it looking for the approximately invariant solution of(1) in the form

119906 = 119908 + 120576V (119905 119909) (48)

Then the invariant equation test119883(119906 minus 119908 minus 120576V)|(48) = 119900(120576)can be written as

1198830 (119906 minus 119908) + 120576 [1198831 (119906 minus 119908) minus 1198830 (V)])119906=119908

= 0 (49)

Note that 1198830 does not contain the differentiation in 119906 there-fore (49) becomes

[1198831 (119906 minus 119908) minus 1198830 (V)]119906=119908

= 0 (50)

whence we obtain the following differential for V(119905 119909)V119905 + V119909 = minus (119908)119909 (51)

Because 119908 = 119908(120577) so (51) can be integrated by the variables

120577 = 119909 minus 1199052

120588 = 119905(52)

Then denoting by1199081015840 the derivative of119908with respect to 120577 wehave

(119908)119905 = minus12 (119908)1015840 (119908)119909 = 1

2 (119908)1015840(53)

and (51) becomes

V120588 = minus12 (119908)1015840 (54)

The integration yields

V = minus12120588 (119908)1015840 + 120595 (119909) (55)

Returning to the variables 119905 119909 we haveV = minus119905 (119908)119909 + 120595 (119909) (56)

Inserting this V in (48) and substituting it into (1) we obtain120595 = ((minus1 minus 3120573)6)119909Thus the approximate symmetry (47) provides the fol-

lowing approximately invariant solution (approximate trav-elling wave)

119906 (119909 119905) = minus12 sdot 1

cosh2 ((119909 minus 119905) 2)+ 120576 [minus 119905

21

cosh2 ((119909 minus 119905) 2) sdot tanh (119909 minus 1199052 )

+ minus1 minus 31205736 119909]

(57)

8 Advances in Mathematical Physics

5 Approximate Conservation Laws

Approximate Lie symmetry can be used to construct theapproximate conservation laws but in this section wewill usepartial Lagrangians to construct approximate conservationlaws of (1) this is a more concise method

Definition 9 (see [10]) An operator 119884 is a kth-order approx-imate Lie-Backlund symmetry

119884 = 1198840 + 1205761198841 + sdot sdot sdot + 120576119896119884119896 (58)

where

119884 = 120585119894 120597120597119909119894 + 120578119894 120597120597119906120572 119894 = 1 2 119899 (59)

and

119884119887 = 120585119894119887 120597120597119909119894 + 120578120572119887 120597

120597119906120572 + 120577120572119887119894 120597120597119906120572119894 + 12057712057211988711989411198942 12059712059711990612057211989411198942 + sdot sdot sdot 119887 = 0 119896

(60)

where 120585119894119887 120578120572119887 isin 119860 and the additional coefficients are

120577120572119887119894 = 119863119894 (119882120572119887 ) + 12058511989511988711990612057211989411989512057712057211988711989411198942 = 11986311989411198631198942 (119882120572119887 ) + 120585119895

11988711990612057211989511989411198942 119894 119895 = 1 119899 (61)

and119882120572119887 is the Lie characteristic function defined by

119882120572119887 = 120578120572119887 minus 120585119895119887119906120572119895 (62)

Definition 10 (see [10]) The approximate Noether operatorassociated with an approximate Lie-Backlund symmetryoperator 119884 is given by

119873119894 = 120585119894 +119882120572 120575120575119906120572 +sum

119904ge1

1198631198941 119863119894s (119882120572) 12057512057511990612057211989411989411198942 119894119904 119894 = 1 119899

(63)

where

119873119894 = 1198731198940 + 1205761198731198941 + sdot sdot sdot + 120576119896119873119894119896 119894 = 1 119899 (64)

where119863119894 is the total derivative operator defined as

119863119894 = 120597120597119909119894 + 119906119894 120597120597119906 + 119906119894119895 120597

120597119906119895 + sdot sdot sdot + 11990611989411989411198942119894119899 12059712059711990611989411198942119894119899

+ sdot sdot sdot 119894 = 1 2 119899(65)

and here119873119894119887 119887 = 0 119896 areNoether operators and the Euler-Lagrange operators are

120575120575119906120572 =

120597120597119906120572 +sum

119904ge1

(minus1)1199041198631198951 119863119895s 12059712059711990612057211989411989511198952119895119904

120572 = 1 119898 119894 = 1 119899(66)

The Euler-Lagrange approximate Lie-Backlund and approx-imate Noether operators are connected by the operatoridentity

119884 + 119863119894 (120585119894) = 119882120572 120575120575119906120572 + 119863119894119873119894 (67)

Definition 11 (see [10]) If there exists a function 119871 = 119871(119909 119906119906(1) 119906(119897) isin 119860 119897 lt 119903 and nonzero functions 119891120573120574 isin 119860 suchthat (1) which can be written as 120575119871120575119906120573 = 120576119891120573120574 1198641205741 where119891120573120574 = 119891120573120574 (119909 119906 119906(1) 119906(119903minus1) 120573 120574 = 1 119898 is an invertiblematrix then provided that 1198641205741 = 0 some 119871 is called a partialLagrangian otherwise it is the standard Lagrangian We termdifferential equations of the form

120575119871120575119906120573 = 120576119891120573120574 1198641205741 (68)

approximate Euler-Lagrange-type equations

Definition 12 (see [5]) An approximate Lie-Backlund sym-metry operator 119884 is called an approximate Noether-typesymmetry operator corresponding to a partial Lagrangian119871 isin119860 if and only if there exists a vector 119861 = (1198611 1198612 119861119899) 119861119894 isin119860 defined by

119861119894 = 1198611198940 + 1205761198611198941 + sdot sdot sdot + 120576119896119861119894119896 (69)

such that

119884 (119871) + 119871119863119894 (120585119894) = 119882120572 120575119871120575119906120572 + 119863119894 (119861119894) + 119900 (120576119896+1) (70)

where 119882 = (11988211198822 119882119898)119882120573 isin 119860 of 119884 is also thecharacteristic of the conservation law 119863119894119879119894 = 119900(120576119896+1) where

119879119894 = 119861119894 minus 119871120585119894 minus119882120573 120597119871120597119906120573119894 + sdot sdot sdot + 119900 (120576119896+1) (71)

of the approximate Euler-Lagrange-type (68)

Because (1) does not have a Lagrange function and if weput a transform 119906 = V119909 then (1) becomes

V119909119905 minus 6V119909V119909119909 + V119909119909119909119909 + 120576120572 (V119909 + 2119909V119909119909)minus 120576120573 (2V119909 + 119909V119909119909) = 0 (72)

In order to write convenient we can get

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 + 120576120572 (119906119909 + 2119909119906119909119909)minus 120576120573 (2119906119909 + 119909119906119909119909) = 0 (73)

Obviously

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 = 0 (74)

and the Lagrange function is

119871 = 12 (119906119909119909)2 + (119906119909)3 minus 1

2119906119909119906119905 (75)

Advances in Mathematical Physics 9

And the approximate 119864119906119897119890119903 minus 119871119886119892119903119886119899119892119890 minus 119905119910119901119890 equation is

120575119871120575119906 = minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909) (76)

So the approximate Noether symmetry operator is 1198840 + 1205981198841for 119871

(1198840 + 1205761198841) 119871 + 119863119894 (1205851198940 + 1205761205851198941) 119871= [(1205780 minus 1205851198950119906119895) + 120576 (1205781 minus 1205851198951119906119895)]sdot [minus120598120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 119863119894 (1198611198940 + 1205761198611198941)

(77)

where

1198840 + 1205761198841 = (12058510 + 12057612058511) 120597120597119905 + (12058520 + 12057612058521) 120597

120597119909 + (1205780+ 1205761205781) 120597

120597119906 + 120577119905 120597120597119906119905 + 120577119909 120597120597119906119909 + 120577119909119909 120597

120597119906119909119909 (78)

120577119905 = 1205780119905 + (1205780119906 minus 12058510119905) 119906119905 minus 12058520119905119906119909 minus 120585101199061199062119905 minus 12058520119906119906119909119906119905+ 12058510119906119905119905 + 12058520119906119909119905 + 120576 [1205781119905 + (1205781119906 minus 12058511119905) 119906119905 minus 12058521119905119906119909minus 120585111199061199062119905 minus 120585211199061199062119909 + 12058511119906119905119905 + 12058521119906119909119905]

(79)

120577119909 = 1205780119909 + (1205780119906 minus 12058520119909) 119906119909 minus 12058510119909119906119905 minus 120585201199061199062119909 minus 12058510119906119906119909119906119905+ 12058510119906119909119905 + 12058520119906119909119909 + 120576 [1205781119909 + (1205781119906 minus 12058521119909) 119906119909 minus 12058511119909119906119905minus 120585211199061199062119909 minus 12058521119906119906119909119906119905 + 12058511119906119909119911119905 + 12058521119906119909119909]

(80)

120577119909119909 = 1205780119909119909 + (21205780119909119906 minus 12058520119909119909) 119906119909 + (1205780119906119906 minus 212058520119909119906) 1199062119909minus 12058510119909119909119906119905 minus 120585101199061199061199062119909119906119905 minus 212058510119909119906119906119909119906119905 minus 12058510119906119906119909119909119906119905minus 212058510119909119906119909119905 minus 212058510119906119906119909119906119909119905 minus 120585201199061199061199063119909 minus 312058520119906119906119909119906119909119909+ (1205780119906 minus 212058520119909) 119906119909119909 + 12058510119906119909119909119905 + 12058520119906119909119909119909 + 120576 [1205781119909119909+ (21205781119909119906 minus 12058521119909119909) 119906119909 + (1205781119906119906 minus 212058521119909119906) 1199062119909 minus 12058511119909119909119906119905minus 120585111199061199061199062119909119906119905 minus 212058511119909119906119906119909119906119905 minus 12058511119906119906119909119909119906119905 minus 212058511119909119906119909119905minus 212058511119906119906119909119906119909119905 minus 120585211199061199061199063119909 minus 312058521119906119906119909119906119909119909+ (1205781119906 minus 212058521119909) 119906119909119909 + 12058511119906119909119909119905 + 12058521119906119909119909119909]

(81)

So (58) becomes

minus 12119906119909120577119905 minus

12119906119905120577119909 + 31199062119909120577119909 + 119906119909119909120577119909119909

+ 119871 [12058510119905 + 12058510119906119906119905 + 120576 (12058511119905 + 12058511119906119906119905)]+ 119871 [12058520119905 + 12058520119906119906119905 + 120576 (12058521119905 + 12058521119906119906119905)]= [1205780 minus 12058510119906119905 minus 12058520119906119909 + 120576 (1205781 minus 12058511119906119905 minus 12058521119906119909)]

sdot [minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 120576 (11986111119905 + 11986111119906119906119905) + 120576 (11986121119905 + 11986121119906119906119909)+ (11986110119905 + 11986110119906119906119905) + (11986120119909 + 11986120119906119906119909)

(82)

Put 119871 120577119905 120577119909 120577119909119909 into (82) and let 1205762 = 0 We obtain

minus121205780119905 minus 11986120119906 = 01205780119909 = 01205780119906 = 012058510 = 012058520 = 0

11986110119906 = 011986120119909 + 11986110119905 = 0

minus121205781119909 minus 11986111119906 = 01205781119909 = 01205781119906 = 012058511 = 012058521 = 0

11986110119906 = 011986121119909 + 11986111119905 = 0

minus121205781119905 + (120572 minus 2120573) 1205780 minus 11986121119906 = 01205781119909119909 + (2120572 minus 120573) 1205780 sdot 119909 = 0

(83)

In the following we will consider three cases of 120572 and 120573First Case 120572 = 2120573 and 120573 = 2120572

12058510 = 012058520 = 01205780 = 012058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = 119891 (119905 119909)

10 Advances in Mathematical Physics

11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905) 1199062 + 119895 (119905 119909)

(84)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909)119891(119905 119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 and 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetric operators

119884 = 120576119892 (119905) 120597120597119906 (85)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12120576119906119909119892 (119905)

1198792 = 119891 (119905 119909) + 120576 [minus12119892119905 (119905) 119906 + 119895 (119905 119909)]minus [120576119892 (119905)] [31199062119909 minus 1

2119906119905] + 119863119909 [120576119892 (119905)] 119906119909119909(86)

Second Case 120572 = 2120573 and 120573 = 212057212058510 = 012058520 = 01205780 = ℎ (119905) 12058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = minusℎ119905 (119905) 1199062 + 119891 (119905 119909) 11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905)2 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)

(87)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909) 119891(119905119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetry operators

119884 = ℎ (119905) 120597120597119906 + 120576119892 (119905) 120597

120597119906 (88)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12119906119909 [ℎ (119905) + 120576119892 (119905)]

1198792 = 119891 (119905 119909) + [minus12ℎ119905 (119905) 119906]+ 120576 [minus12119892119905 (119905) 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)]minus [120576119892 (119905) + ℎ (119905)] [31199062119909 minus 1

2119906119905]+ 119863119909 [120576119892 (119905) + ℎ (119905)] 119906119909119909

(89)

Third Case 120572 = 2120573 and 120573 = 2120572 the conservation laws are thesame as the first case

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Thisworkwas supported in part by theNatural Science Foun-dation of Inner Mongolia of China under grant 2016MS0116

References

[1] A Wazwaz Partial Differential Equations and Solitary WavesTheory Higher Education Press Beijing China 2009

[2] Y S Kivshar and B A Malomed ldquoDynamics of solitons innearly integrable systemsrdquo Reviews of Modern Physics vol 61no 4 pp 763ndash915 1989

[3] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMatematicheskii Sbornik vol 64 no 2 p 427 1989

[4] V A Bailov R K Gazizov and N H Ibragmov ldquoPertubationmethods in group analysisrdquo Journal of Mathematical Sciencesvol 55 no 1 pp 1450ndash1485 1991

[5] N H Ibragimov and V F Kovalev Approximate and Renor-mgroup Symmetries Higher Education Press Beijing China2009

[6] V A Baikov and N H Ibragimov ldquoContinuation of approxi-mate transformation groups via multiple time scales methodrdquoNonlinear Dynamics vol 22 no 1 pp 3ndash13 2000

[7] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations vol 168 ofApplied Mathematical Sciences Springer New York NY USA2010

[9] A H Kara F M Mahomed and G Unal ldquoApproximate sym-metries and conservation laws with applicationsrdquo InternationalJournal ofTheoretical Physics vol 38 no 9 pp 2389ndash2399 1999

[10] A G Johnpillai A H Kara and FMMahomed ldquoApproximateNoether-type symmetries and conservation laws via partial

Advances in Mathematical Physics 11

Lagrangians for PDEs with a small parameterrdquo Journal of Com-putational and AppliedMathematics vol 223 no 1 pp 508ndash5182009

[11] J-R Su S-L Zhang and J-N Li ldquoApproximate noether-typesymmetries and conservation laws via partial lagrangians fornonlinear wave equation with dampingrdquo Communications inTheoretical Physics vol 53 no 1 pp 37ndash42 2010

[12] E S Benilov and B A Malomed ldquoOn the spectral singularityproblem in the perturbation theory for systems close to theKdVequationrdquo Journal of Physics A Mathematical and General vol21 no 8 pp L439ndashL442 1988

[13] M A Allen and G Rowlands ldquoA solitary-wave solution to aperturbedKdV equationrdquo Journal of Plasma Physics vol 64 no4 pp 475ndash480 2000

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Advances in Mathematical Physics 3

for the exact symmetries of the unperturbed equation 119906119905 minus6119906119906119909 + 119906119909119909119909 = 0 we can get

1205780119905 = 01205780119909 = 01205910119909 = 01205910119906 = 0

1205780119906 + 21205850119909 = 01205850119909119909 = 01205850119906 = 0

61205780 + 1205850119905 + 121199061205850119909 = 031205850119909 minus 1205910119905 = 0

(14)

Then we obtain 1205850 = 1198881119909 minus 61198882119905 + 1198883 1205910 = 31198881119905 + 1198882 1205780 =minus21198881119906 + 1198882 where 1198881 1198882 1198883 1198884 are arbitrary constants Hence1198830 = (1198881119909 minus 61198882119905 + 1198883) 120597

120597119909 + (31198881119905 + 1198884) 120597120597119905

+ (minus21198881119906 + 1198882) 120597120597119906

(15)

In other words 119906119905 minus 6119906119906119909 + 119906119909119909119909 = 0 admits the four-dimen-sional Lie algebra with the basis

11988310 = 119909 120597120597119909 + 3119905 120597120597119905 minus 2119906 120597

120597119906 11988320 = minus6119905 120597

120597119909 + 120597120597119906

11988330 = 120597120597119909

11988340 = 120597120597119906

(16)

32 Approximate Symmetries and Optimal System

321 Approximate Symmetries The Cases 120572 and 120573 Are Arbi-trary First we need to determine the auxiliary function 119867by virtue of (15) by

119867 = 1120576 [119883(119896)0 (1198650 (119911) + 1205761198651 (119911))100381610038161003816100381610038161198650(119911)+1205761198651(119911)=0] (17)

Substituting the expression (15) of the generator 1198830 intoabove equation we obtain the auxiliary function

119867 = 1198882120572 minus 21198882120573 + (31198881120572 minus 61198881120573) 119906+ (21198883120572 minus 121198882120572119905 + 61198881120572119909 minus 1198883120573 + 61198882120573119905 minus 31198881120573119909)sdot 119906119909

(18)

Now calculate the operators 1198831 by solving the inhomoge-neous determining equation for deformations

119883(119896)1 1198650 (119911)100381610038161003816100381610038161198650(119911)=0 + 119867 = 0 (19)

Above determining equation yields

1198882120572 minus 21198882120573 minus 61198881120572119906 minus 61198881120573119906 + 1205781119905 = 021198881120572 minus 21205781119909 = 01205911119909 = 01205781119906 = 01205781119906 + 21205851119909 = 01205851119909119909 = 01205911119906 = 031205851119909 minus 1205911119905 = 021198883120572 minus 1198883120573 + 61198881120572119909 minus 31198881120573119909 minus 121198882120572119905 + 61198882120573119905 minus 61205781

minus 1205851119905 minus 121199061205851119909 = 0

(20)

Solving this system we obtain

1198831 = [minus3 (1198881120572 + 1198881120573) 119905119909 + (21198883120572 minus 1198883120573 minus 61198891) 119905minus 3 (1198882120572 + 1198882120573) 1199052 + 1198892] 120597

120597119909+ [minus9 (1198881120572 + 1198881120573) 11990522 + 1198893] 120597

120597119905 + [311988811205721199092+ (21198882120573 + 61198881120572119906 minus 1198882120572 + 61198881120573119906) 119905 + 1198891] 120597

120597119906

(21)

Then we obtain the following approximate symmetries of (1)

119883 = [1198881119909 minus 61198882119905 + 1198883 + 120576 [minus3 (1198881120572 + 1198881120573) 119905119909+ (21198883120572 minus 1198883120573 minus 61198891) 119905 minus 3 (1198882120572 + 1198882120573) 1199052

+ 1198892]] 120597120597119909 + [31198881119905 + 1198882 + 120576 [minus9 (1198881120572 + 1198881120573) 11990522

+ 1198893]] 120597120597119905 + [minus21198881119906 + 1198882 + 120576 [311988811205721199092

+ (21198882120573 + 61198881120572119906 minus 1198882120572 + 61198881120573119906) 119905 + 1198891]] 120597120597119906

(22)

and we have

V1 = 119909 120597120597119909 + 3119905 120597120597119905 minus 2119906 120597

120597119906 + 120576 [minus3 (120572 + 120573) 119905119909 120597120597119909

minus 92 (120572 + 120573) 1199052 120597120597119905 + (31205721199092 + 6120572119906119905 + 6120573119906119905) 120597

120597119906]

V2 = minus6119905 120597120597119909 + 120597

120597119906 + 120576 [minus3 (120572 + 120573) 1199052 120597120597119909+ (2120573 minus 120572) 119905 120597

120597119906]

V3 = 120597120597119909 + 120576 [(2120572 minus 120573) 119905 120597

120597119909]

4 Advances in Mathematical Physics

V4 = 120597120597119906

V5 = 120576V2V6 = 120576V3V7 = 120576V4

(23)

Tables 1 2 and 3 of commutator evaluated in the first-order of precision show that above operators span a seven-dimensional approximate Lie algebra 1198717322 Approximate Symmetries The Case 120572 = 2120573 When 120572 =2120573 the equation admits seven-dimensional approximate Liealgebra 1198717 as follows

V1 = 119909 120597120597119909 + 3119905 120597120597119905 minus 2119906 120597

120597119906 + 120576 [minus9120572119905119909 120597120597119909 minus 27

2 1205721199052 120597120597119905+ (31205721199092 + 6120572119906119905 + 6120573119906119905) 120597

120597119906]

V2 = minus6119905 120597120597119909 + 120597120597119906 + 120576 [minus91205721199052 120597120597119909]

V3 = 120597120597119909 + 120576 [3120572119905 120597

120597119909] V4 = 120597

120597119906V5 = 120576V2V6 = 120576V3V7 = 120576V4

(24)

323 Approximate Symmetries and Optimal System The Case120572 = minus120573 When 120572 = minus120573 the equation admits seven-dimen-sional approximate Lie algebra 1198717 as follows

V1 = 119909 120597120597119909 + 3119905 120597120597119905 minus 2119906 120597

120597119906 + 120576 [31205721199092120597120597119906]

V2 = minus6119905 120597120597119909 + 120597120597119906 + 120576 [minus3120572119905 120597120597119906]

V3 = 120597120597119909 + 120576 [3120572119905 120597

120597119909] V4 = 120597

120597119906V5 = 120576V2V6 = 120576V3V7 = 120576V4

(25)

It is worth nothing that the seven-dimensional approxi-mate Lie algebra 1198717 = 119892 is solvable and its finite sequence ofideals is as follows

0 sub ⟨V6⟩ sub ⟨V5 V6⟩ sub ⟨V3 V5 V6⟩ sub ⟨V2 V3 V5 V6⟩sub ⟨V2 V3 V4 V5 V6⟩ sub ⟨V2 V3 V4 V5 V6 V7⟩ sub 119892 (26)

In the following we will construct the optimal system ofabove Lie algebra 1198717Themethod used here for obtaining theone-dimensional optimal system of subalgebras is that givenin [7] This approach is taking a general element from theLie algebra and reducing it to its simplest equivalent formby applying carefully chosen adjoint transformations that aredefined as follows

Definition 6 (see [7]) Let119866 be a Lie groupAnoptimal systemof s-parameter subgroups is a list of conjugacy inequivalents-parameter subalgebras with the property that any othersubgroup is conjugate to precisely one subgroup in the listSimilarly a list of s-parameter subalgebras forms an optimalsystem if every s-parameter subalgebra of 119892 is equivalent to aunique member of the list under some element of the adjointrepresentation ℎ = 119860119889(119892(ℎ)) 119892 isin 119866Theorem7 (see [7]) Let119867 and be connected s-dimensionalLie subgroups of the Lie group 119866 with corresponding Liesubalgebras ℎ and ℎ of the Lie algebra 119892 of119866Then119867 = 119892119867119892minus1are conjugate subgroups if and only if ℎ = 119860119889(119903(ℎ)) are con-jugate subalgebras

By Theorem 7 the problem of finding an optimal systemof subgroups is equivalent to that of finding an optimal systemof subalgebras To compute the adjoint representation we usethe Lie series

119860119889 (exp (120583V119894)) V119895 = V119895 minus 120583 [V119894 V119895] + 12058322 [V119894 [V119894 V119895]]

minus sdot sdot sdot = exp (119886119889 (minus120583V119894)) V119895(27)

where [V119894 V119895] is the commutator for the Lie algebra and 120583 isa parameterand 119894 119895 = 1 7 In this manner we constructTable 4 with the (119894 119895)th entry indicating 119860119889(exp(120583V119894))V119895Theorem 8 An optimal system of one-dimensional approx-imate symmetry algebra (case 120572 = minus120573) of equation (1) isprovided by V7 V6 V1 + 119887V6 V3 + 119887V7 V2 + 119887V5 + 119888V7 V5 + 119887V3 +119888V7 V4 + 119887V2 + 119888V5 + 119889V7Proof Considering the approximate symmetry algebra g of(1) whose adjoint representation was determined in theTable 4 our task is to simplify as many of the coefficients 119886119894 aspossible through judicious applications of adjoint maps to119881119894so that 119881119894 is equivalent 1198811015840119894 under the adjoint representation

Given a nonzero vector

1198811 = 1198861V1 + 1198862V2 + 1198863V3 + 1198864V4 + 1198865V5 + 1198866V6 + 1198867V7 (28)

Advances in Mathematical Physics 5

Table 1 Approximate commutators of approximate symmetry algebra of (1) (119886 = 2120572 minus 120573 119887 = 2120573 minus 120572 119888 = 120572 + 120573 119889 = 31205722)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus119889V5 minus V3 minus3V4 + 3120576119888V1 2V5 minusV6 minus3V7V2 minus2V2 0 0 6V3 minus 119887V5 0 0 6V6V3 119889V5 + V3 0 0 minus119886V6 0 0 0V4 3V4 minus 3120576119888V1 minus6V3 + 119887V5 119886V6 0 minus6V6 0 0V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 2 Approximate commutators of approximate symmetry algebra of (1) (120572 = 2120573 119886 = 3120572)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus1198862 V5 minus V3 minus3V4 + 3120576119886V1 2V5 minusV6 minus3V7

V2 minus2V2 0 0 minus6V3 + 52 V5 0 0 6V6

V3 V3 + 52V5 0 0 minus119886V6 0 0 0

V4 3V4 minus 3120576119886V1 6V3 minus 52 V5 119886V6 0 minus6V6 0 0

V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 3 Approximate commutators of approximate symmetry algebra of (1) ( 120572 = minus120573 119886 = 3120572)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus1198862 V5 minus V3 minus3V4 2V5 minusV6 minus3V7

V2 minus2V2 0 0 6V3 + 119886V5 0 0 6V6V3

1198862 V5 + V3 0 0 minus119886V6 0 0 0

V4 3V4 minus6V3 minus 119886V5 119886V6 0 minus6V6 0 0V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 4 Adjoint representation of approximate symmetry of (1)

119860119889 V1 V2 V3 V4 V5 V6 V7V1 V1 119890minus2120583V2 119890minus120583V3 + 119886

2 119890minus120583V5 minus1198862 V5 1198903120583V4 119890minus2120583V5 119890minus120583V6 1198903120583V7

V2 V1 + 2120583V2 V2 V3 V4 minus 120583(6V3 + 119886V5) V5 V6 V7 minus 6120583V6V3 V1 minus 120583(119886V52 + V3) V2 V3 V4 + 120583(119886V6) V5 V6 V7V4 V1 minus 3120583V4 V2 + 120583(6V3 + 119886V5) V3 minus 119886120583V6 V4 V5 + 6120583V6 V6 V7V5 V1 + 2120583V5 V2 V3 V4 minus 6120583V6 V5 V6 V7V6 V1 minus 120583V6 V2 V3 V4 V5 V6 V7V7 V1 minus 3120583V7 V2 + 6120583V7 V3 V4 V5 V6 V7

First suppose that 1198861 = 0 Scaling 1198811 if necessary we canassume that 1198861 = 1 As for the 7th column of the Table 4we have

11988110158401 = 119860119889(exp ((11988643 V4))) ∘ 119860119889(exp ((minus11988622 V2)))∘ 119860119889 (exp ((11988673 V7)))

∘ 119860119889 (exp ((1198863 + 211988621198864) V3)))∘ 119860119889 (exp ((minus11988652 + 11988611988634 + 1198861198862119886412 ) V5)))1198811

= V1 + (1198866 minus 119886119886311988643 + 211988641198865 + 311988621198867) V6(29)

6 Advances in Mathematical Physics

The remaining approximate one-dimensional subalgebras arespanned by vectors of the above form with 1198861 = 0 If 1198864 = 0we have

1198812 = 1198862V2 + 1198863V3 + V4 + 1198865V5 + 1198866V6 + 1198867V7 (30)

Next we act on 1198812 to cancel the coefficients of V3 and V6 asfollows

11988110158402 = 119860119889(exp((minus119886361198862 V4)))

∘ 119860119889 (exp((minus119886561198862 +11988611988623361198862 +

11988666 ) V5)))1198812= 1198862V2 + V4 + (1198865 minus 11988611988636 V5 + 1198867V7

(31)

If 1198861 1198864 = 0 and 1198862 = 0 the nonzero vector1198813 = V2 + 1198863V3 + 1198865V5 + 1198866V6 + 1198867V7 (32)

is equivalent to

11988110158403 = 119860119889(exp ((minus11988636 V4)))

∘ 119860119889 (exp((minus11988666 + minus1198861198862336 + 119886611988656 ) V7)))1198813= V2 + (1198865 minus 11988611988636 ) V5 + 1198867V7

(33)

If 1198861 1198862 1198864 = 0 and 1198865 = 0 we scale to make 1198865 = 1 and then

1198814 = 1198863V3 + V5 + 1198866V6 + 1198867V7 (34)

is equivalent to 11988110158404 under the adjoint representation11988110158404 = 119860119889(exp(( 119886661198867 V2)))1198814 = 1198863V3 + V5 + 1198867V7 (35)

If 1198861 1198862 1198864 1198865 = 0 and 1198863 = 0 in the same way as before thenonzero vector

1198815 = V3 + 1198866V6 + 1198867V7 (36)

can be simplified as

11988110158405 = 119860119889(exp(( 119886661198867 V2)))1198812 = V3 + 1198867V7 (37)

If 1198861 1198862 1198863 1198864 1198865 = 0 and 1198867 = 0 we can act

1198816 = 1198866V6 + V7 (38)

by119860119889(exp(((11988666)V2))) to cancel the coefficient of V6 leadingto

11988110158406 = 119860119889 (exp ((11988666 V2)))1198816 = V7 (39)

The last remaining case occurs when 1198861 1198862 1198863 1198864 1198865 1198867 =0 and 1198866 = 0 for which our earlier simplifications wereunnecessary because the only remaining vectors are themultiples of 1198866 on which the adjoint representation actstrivially

4 Approximately Invariant Solutions

In this section we use two different techniques to constructnew approximate solutions of (1) when 120572 = minus12057341 Approximately Invariant Solutions I In the beginning ofthis section we compute an approximately invariant solutionbased on the 119883 = V2 The approximate invariants for 119883 aredetermined by

119883 (119869) = (minus6119905 120597120597119909 + 120597

120597119906 + 120576 (minus3120572119905 120597120597119906)) (1198690 + 1205761198691)= 119900 (120576)

(40)

Equivalently

(minus6119905 120597120597119909 + 120597120597119906) (1198690) = 0

(minus3120572119905 120597120597119906) (1198691) + (minus6119905 120597120597119909 + 120597120597119906) (1198690) = 0

(41)

Thefirst equation has two functionally independent solutions1198690 = 119905 and 1198690 = 1199096119905 + 119906 The simplest solutions of thesecond equation are respectively 1198691 = 0 and 1198691 = minus1205721199092 + 119905Therefore we have two independent invariants to 1199096119905 + 119906 +120576(minus1205721199092 + 119905) and 120593(119905) with respect to119883

Letting 1199096119905 + 119906 + 120576(3120572119905119906) = 120593(119905) we obtain 119906 = 120593(119905) minus1199096119905minus120576(minus1205721199092+ 119905) for the approximately invariant solutionsTherefore we can obtain

119906 (119909 119905) asymp minus 1199096119905 + Γ0 (119905) + 120576 (minus119905 minus 120572119909

2 + Γ1 (119905)+ Γ0 (119905) Γ10158401 (119905)

(42)

Putting (42) into (1) we obtain

Γ0 (119905) = 1198621119879Γ1 (119905) = 11986221199053 + 119905 + 1198622119905

(43)

where 1198621 and 1198622 are arbitrary constants Hence we obtainedthe approximately invariant solution to (1)

119906 (119909 119905) asymp minus 1199096119905 +

1198621119905 + 120576 (1198622119905 minus 1205721199092 ) (44)

In this manner we compute approximate invariants withrespect to the generators of Lie algebra and optimal systemas shown in Table 5

42 Approximately Invariant Solution II Now we apply adifferent technique to find approximately invariant solutionsfor (1) We will begin with one exact solution

119906 = 119908 (120577) = minus12

1cosh2120577 (45)

Advances in Mathematical Physics 7

Table 5 Approximate invariants with respect to the generators of Lie algebra and optimal system

Operator Approximate Invariants

V11199051199093 1199092119906 minus 120576(1205721199093

2 )V2 119905 1199096119905 + 119906 + 120576 (minus1205721199092 + 119905)V3 119905 119906V4 119909 119906V5 119905 1199096119905 + 119906V6 119905 119906V7 119909 119906V1 + 119887V6 119905

1199093 + 120576(119887119909minus4119905) 1199092119906 minus 120576(12057211990932 + 2119887119909

3119906 )V3 + 119887V7 119905 minus 120576(119887119909) 119906V4 + 119887V2 + 119888V5 + 119889V7 1199052

2 + 1199096119887 + 120576 (( 119888119887 minus 119889) 119905) 119887119906 minus 119905 + 120576(31205721199052

2 + 119889119905)V2 + 119887V5 + 119888V7 119905 minus 120576 (119906119888 )

1199096119905 + 119906 + 120576(3120572119905119906 minus 1198881199092

721199053 )V5 + 119887V3 + 119888V7 119905 minus 120576 (119888119909119887 ) 119906

of the unperturbed equation 119906119905 minus 6119906119906119909 + 119906119909119909119909 = 0 Here 120577 =(119909 minus 119905)2 is an invariant of the group with the generator

1198830 = 120597120597119905 +

120597120597119909 (46)

The function 119908 given by (45) is invariant under the operator(46)

Using the generators 1198831 = 120597120597119909 admitted by 119906119905 minus 6119906119906119909 +119906119909119909119909 = 0 we will take the approximate symmetry

119883 = 1198830 + 1205761198831 = 120597120597119905 +

120597120597119909 + 120576 120597

120597119909 (47)

and use it looking for the approximately invariant solution of(1) in the form

119906 = 119908 + 120576V (119905 119909) (48)

Then the invariant equation test119883(119906 minus 119908 minus 120576V)|(48) = 119900(120576)can be written as

1198830 (119906 minus 119908) + 120576 [1198831 (119906 minus 119908) minus 1198830 (V)])119906=119908

= 0 (49)

Note that 1198830 does not contain the differentiation in 119906 there-fore (49) becomes

[1198831 (119906 minus 119908) minus 1198830 (V)]119906=119908

= 0 (50)

whence we obtain the following differential for V(119905 119909)V119905 + V119909 = minus (119908)119909 (51)

Because 119908 = 119908(120577) so (51) can be integrated by the variables

120577 = 119909 minus 1199052

120588 = 119905(52)

Then denoting by1199081015840 the derivative of119908with respect to 120577 wehave

(119908)119905 = minus12 (119908)1015840 (119908)119909 = 1

2 (119908)1015840(53)

and (51) becomes

V120588 = minus12 (119908)1015840 (54)

The integration yields

V = minus12120588 (119908)1015840 + 120595 (119909) (55)

Returning to the variables 119905 119909 we haveV = minus119905 (119908)119909 + 120595 (119909) (56)

Inserting this V in (48) and substituting it into (1) we obtain120595 = ((minus1 minus 3120573)6)119909Thus the approximate symmetry (47) provides the fol-

lowing approximately invariant solution (approximate trav-elling wave)

119906 (119909 119905) = minus12 sdot 1

cosh2 ((119909 minus 119905) 2)+ 120576 [minus 119905

21

cosh2 ((119909 minus 119905) 2) sdot tanh (119909 minus 1199052 )

+ minus1 minus 31205736 119909]

(57)

8 Advances in Mathematical Physics

5 Approximate Conservation Laws

Approximate Lie symmetry can be used to construct theapproximate conservation laws but in this section wewill usepartial Lagrangians to construct approximate conservationlaws of (1) this is a more concise method

Definition 9 (see [10]) An operator 119884 is a kth-order approx-imate Lie-Backlund symmetry

119884 = 1198840 + 1205761198841 + sdot sdot sdot + 120576119896119884119896 (58)

where

119884 = 120585119894 120597120597119909119894 + 120578119894 120597120597119906120572 119894 = 1 2 119899 (59)

and

119884119887 = 120585119894119887 120597120597119909119894 + 120578120572119887 120597

120597119906120572 + 120577120572119887119894 120597120597119906120572119894 + 12057712057211988711989411198942 12059712059711990612057211989411198942 + sdot sdot sdot 119887 = 0 119896

(60)

where 120585119894119887 120578120572119887 isin 119860 and the additional coefficients are

120577120572119887119894 = 119863119894 (119882120572119887 ) + 12058511989511988711990612057211989411989512057712057211988711989411198942 = 11986311989411198631198942 (119882120572119887 ) + 120585119895

11988711990612057211989511989411198942 119894 119895 = 1 119899 (61)

and119882120572119887 is the Lie characteristic function defined by

119882120572119887 = 120578120572119887 minus 120585119895119887119906120572119895 (62)

Definition 10 (see [10]) The approximate Noether operatorassociated with an approximate Lie-Backlund symmetryoperator 119884 is given by

119873119894 = 120585119894 +119882120572 120575120575119906120572 +sum

119904ge1

1198631198941 119863119894s (119882120572) 12057512057511990612057211989411989411198942 119894119904 119894 = 1 119899

(63)

where

119873119894 = 1198731198940 + 1205761198731198941 + sdot sdot sdot + 120576119896119873119894119896 119894 = 1 119899 (64)

where119863119894 is the total derivative operator defined as

119863119894 = 120597120597119909119894 + 119906119894 120597120597119906 + 119906119894119895 120597

120597119906119895 + sdot sdot sdot + 11990611989411989411198942119894119899 12059712059711990611989411198942119894119899

+ sdot sdot sdot 119894 = 1 2 119899(65)

and here119873119894119887 119887 = 0 119896 areNoether operators and the Euler-Lagrange operators are

120575120575119906120572 =

120597120597119906120572 +sum

119904ge1

(minus1)1199041198631198951 119863119895s 12059712059711990612057211989411989511198952119895119904

120572 = 1 119898 119894 = 1 119899(66)

The Euler-Lagrange approximate Lie-Backlund and approx-imate Noether operators are connected by the operatoridentity

119884 + 119863119894 (120585119894) = 119882120572 120575120575119906120572 + 119863119894119873119894 (67)

Definition 11 (see [10]) If there exists a function 119871 = 119871(119909 119906119906(1) 119906(119897) isin 119860 119897 lt 119903 and nonzero functions 119891120573120574 isin 119860 suchthat (1) which can be written as 120575119871120575119906120573 = 120576119891120573120574 1198641205741 where119891120573120574 = 119891120573120574 (119909 119906 119906(1) 119906(119903minus1) 120573 120574 = 1 119898 is an invertiblematrix then provided that 1198641205741 = 0 some 119871 is called a partialLagrangian otherwise it is the standard Lagrangian We termdifferential equations of the form

120575119871120575119906120573 = 120576119891120573120574 1198641205741 (68)

approximate Euler-Lagrange-type equations

Definition 12 (see [5]) An approximate Lie-Backlund sym-metry operator 119884 is called an approximate Noether-typesymmetry operator corresponding to a partial Lagrangian119871 isin119860 if and only if there exists a vector 119861 = (1198611 1198612 119861119899) 119861119894 isin119860 defined by

119861119894 = 1198611198940 + 1205761198611198941 + sdot sdot sdot + 120576119896119861119894119896 (69)

such that

119884 (119871) + 119871119863119894 (120585119894) = 119882120572 120575119871120575119906120572 + 119863119894 (119861119894) + 119900 (120576119896+1) (70)

where 119882 = (11988211198822 119882119898)119882120573 isin 119860 of 119884 is also thecharacteristic of the conservation law 119863119894119879119894 = 119900(120576119896+1) where

119879119894 = 119861119894 minus 119871120585119894 minus119882120573 120597119871120597119906120573119894 + sdot sdot sdot + 119900 (120576119896+1) (71)

of the approximate Euler-Lagrange-type (68)

Because (1) does not have a Lagrange function and if weput a transform 119906 = V119909 then (1) becomes

V119909119905 minus 6V119909V119909119909 + V119909119909119909119909 + 120576120572 (V119909 + 2119909V119909119909)minus 120576120573 (2V119909 + 119909V119909119909) = 0 (72)

In order to write convenient we can get

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 + 120576120572 (119906119909 + 2119909119906119909119909)minus 120576120573 (2119906119909 + 119909119906119909119909) = 0 (73)

Obviously

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 = 0 (74)

and the Lagrange function is

119871 = 12 (119906119909119909)2 + (119906119909)3 minus 1

2119906119909119906119905 (75)

Advances in Mathematical Physics 9

And the approximate 119864119906119897119890119903 minus 119871119886119892119903119886119899119892119890 minus 119905119910119901119890 equation is

120575119871120575119906 = minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909) (76)

So the approximate Noether symmetry operator is 1198840 + 1205981198841for 119871

(1198840 + 1205761198841) 119871 + 119863119894 (1205851198940 + 1205761205851198941) 119871= [(1205780 minus 1205851198950119906119895) + 120576 (1205781 minus 1205851198951119906119895)]sdot [minus120598120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 119863119894 (1198611198940 + 1205761198611198941)

(77)

where

1198840 + 1205761198841 = (12058510 + 12057612058511) 120597120597119905 + (12058520 + 12057612058521) 120597

120597119909 + (1205780+ 1205761205781) 120597

120597119906 + 120577119905 120597120597119906119905 + 120577119909 120597120597119906119909 + 120577119909119909 120597

120597119906119909119909 (78)

120577119905 = 1205780119905 + (1205780119906 minus 12058510119905) 119906119905 minus 12058520119905119906119909 minus 120585101199061199062119905 minus 12058520119906119906119909119906119905+ 12058510119906119905119905 + 12058520119906119909119905 + 120576 [1205781119905 + (1205781119906 minus 12058511119905) 119906119905 minus 12058521119905119906119909minus 120585111199061199062119905 minus 120585211199061199062119909 + 12058511119906119905119905 + 12058521119906119909119905]

(79)

120577119909 = 1205780119909 + (1205780119906 minus 12058520119909) 119906119909 minus 12058510119909119906119905 minus 120585201199061199062119909 minus 12058510119906119906119909119906119905+ 12058510119906119909119905 + 12058520119906119909119909 + 120576 [1205781119909 + (1205781119906 minus 12058521119909) 119906119909 minus 12058511119909119906119905minus 120585211199061199062119909 minus 12058521119906119906119909119906119905 + 12058511119906119909119911119905 + 12058521119906119909119909]

(80)

120577119909119909 = 1205780119909119909 + (21205780119909119906 minus 12058520119909119909) 119906119909 + (1205780119906119906 minus 212058520119909119906) 1199062119909minus 12058510119909119909119906119905 minus 120585101199061199061199062119909119906119905 minus 212058510119909119906119906119909119906119905 minus 12058510119906119906119909119909119906119905minus 212058510119909119906119909119905 minus 212058510119906119906119909119906119909119905 minus 120585201199061199061199063119909 minus 312058520119906119906119909119906119909119909+ (1205780119906 minus 212058520119909) 119906119909119909 + 12058510119906119909119909119905 + 12058520119906119909119909119909 + 120576 [1205781119909119909+ (21205781119909119906 minus 12058521119909119909) 119906119909 + (1205781119906119906 minus 212058521119909119906) 1199062119909 minus 12058511119909119909119906119905minus 120585111199061199061199062119909119906119905 minus 212058511119909119906119906119909119906119905 minus 12058511119906119906119909119909119906119905 minus 212058511119909119906119909119905minus 212058511119906119906119909119906119909119905 minus 120585211199061199061199063119909 minus 312058521119906119906119909119906119909119909+ (1205781119906 minus 212058521119909) 119906119909119909 + 12058511119906119909119909119905 + 12058521119906119909119909119909]

(81)

So (58) becomes

minus 12119906119909120577119905 minus

12119906119905120577119909 + 31199062119909120577119909 + 119906119909119909120577119909119909

+ 119871 [12058510119905 + 12058510119906119906119905 + 120576 (12058511119905 + 12058511119906119906119905)]+ 119871 [12058520119905 + 12058520119906119906119905 + 120576 (12058521119905 + 12058521119906119906119905)]= [1205780 minus 12058510119906119905 minus 12058520119906119909 + 120576 (1205781 minus 12058511119906119905 minus 12058521119906119909)]

sdot [minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 120576 (11986111119905 + 11986111119906119906119905) + 120576 (11986121119905 + 11986121119906119906119909)+ (11986110119905 + 11986110119906119906119905) + (11986120119909 + 11986120119906119906119909)

(82)

Put 119871 120577119905 120577119909 120577119909119909 into (82) and let 1205762 = 0 We obtain

minus121205780119905 minus 11986120119906 = 01205780119909 = 01205780119906 = 012058510 = 012058520 = 0

11986110119906 = 011986120119909 + 11986110119905 = 0

minus121205781119909 minus 11986111119906 = 01205781119909 = 01205781119906 = 012058511 = 012058521 = 0

11986110119906 = 011986121119909 + 11986111119905 = 0

minus121205781119905 + (120572 minus 2120573) 1205780 minus 11986121119906 = 01205781119909119909 + (2120572 minus 120573) 1205780 sdot 119909 = 0

(83)

In the following we will consider three cases of 120572 and 120573First Case 120572 = 2120573 and 120573 = 2120572

12058510 = 012058520 = 01205780 = 012058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = 119891 (119905 119909)

10 Advances in Mathematical Physics

11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905) 1199062 + 119895 (119905 119909)

(84)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909)119891(119905 119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 and 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetric operators

119884 = 120576119892 (119905) 120597120597119906 (85)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12120576119906119909119892 (119905)

1198792 = 119891 (119905 119909) + 120576 [minus12119892119905 (119905) 119906 + 119895 (119905 119909)]minus [120576119892 (119905)] [31199062119909 minus 1

2119906119905] + 119863119909 [120576119892 (119905)] 119906119909119909(86)

Second Case 120572 = 2120573 and 120573 = 212057212058510 = 012058520 = 01205780 = ℎ (119905) 12058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = minusℎ119905 (119905) 1199062 + 119891 (119905 119909) 11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905)2 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)

(87)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909) 119891(119905119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetry operators

119884 = ℎ (119905) 120597120597119906 + 120576119892 (119905) 120597

120597119906 (88)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12119906119909 [ℎ (119905) + 120576119892 (119905)]

1198792 = 119891 (119905 119909) + [minus12ℎ119905 (119905) 119906]+ 120576 [minus12119892119905 (119905) 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)]minus [120576119892 (119905) + ℎ (119905)] [31199062119909 minus 1

2119906119905]+ 119863119909 [120576119892 (119905) + ℎ (119905)] 119906119909119909

(89)

Third Case 120572 = 2120573 and 120573 = 2120572 the conservation laws are thesame as the first case

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Thisworkwas supported in part by theNatural Science Foun-dation of Inner Mongolia of China under grant 2016MS0116

References

[1] A Wazwaz Partial Differential Equations and Solitary WavesTheory Higher Education Press Beijing China 2009

[2] Y S Kivshar and B A Malomed ldquoDynamics of solitons innearly integrable systemsrdquo Reviews of Modern Physics vol 61no 4 pp 763ndash915 1989

[3] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMatematicheskii Sbornik vol 64 no 2 p 427 1989

[4] V A Bailov R K Gazizov and N H Ibragmov ldquoPertubationmethods in group analysisrdquo Journal of Mathematical Sciencesvol 55 no 1 pp 1450ndash1485 1991

[5] N H Ibragimov and V F Kovalev Approximate and Renor-mgroup Symmetries Higher Education Press Beijing China2009

[6] V A Baikov and N H Ibragimov ldquoContinuation of approxi-mate transformation groups via multiple time scales methodrdquoNonlinear Dynamics vol 22 no 1 pp 3ndash13 2000

[7] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations vol 168 ofApplied Mathematical Sciences Springer New York NY USA2010

[9] A H Kara F M Mahomed and G Unal ldquoApproximate sym-metries and conservation laws with applicationsrdquo InternationalJournal ofTheoretical Physics vol 38 no 9 pp 2389ndash2399 1999

[10] A G Johnpillai A H Kara and FMMahomed ldquoApproximateNoether-type symmetries and conservation laws via partial

Advances in Mathematical Physics 11

Lagrangians for PDEs with a small parameterrdquo Journal of Com-putational and AppliedMathematics vol 223 no 1 pp 508ndash5182009

[11] J-R Su S-L Zhang and J-N Li ldquoApproximate noether-typesymmetries and conservation laws via partial lagrangians fornonlinear wave equation with dampingrdquo Communications inTheoretical Physics vol 53 no 1 pp 37ndash42 2010

[12] E S Benilov and B A Malomed ldquoOn the spectral singularityproblem in the perturbation theory for systems close to theKdVequationrdquo Journal of Physics A Mathematical and General vol21 no 8 pp L439ndashL442 1988

[13] M A Allen and G Rowlands ldquoA solitary-wave solution to aperturbedKdV equationrdquo Journal of Plasma Physics vol 64 no4 pp 475ndash480 2000

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4 Advances in Mathematical Physics

V4 = 120597120597119906

V5 = 120576V2V6 = 120576V3V7 = 120576V4

(23)

Tables 1 2 and 3 of commutator evaluated in the first-order of precision show that above operators span a seven-dimensional approximate Lie algebra 1198717322 Approximate Symmetries The Case 120572 = 2120573 When 120572 =2120573 the equation admits seven-dimensional approximate Liealgebra 1198717 as follows

V1 = 119909 120597120597119909 + 3119905 120597120597119905 minus 2119906 120597

120597119906 + 120576 [minus9120572119905119909 120597120597119909 minus 27

2 1205721199052 120597120597119905+ (31205721199092 + 6120572119906119905 + 6120573119906119905) 120597

120597119906]

V2 = minus6119905 120597120597119909 + 120597120597119906 + 120576 [minus91205721199052 120597120597119909]

V3 = 120597120597119909 + 120576 [3120572119905 120597

120597119909] V4 = 120597

120597119906V5 = 120576V2V6 = 120576V3V7 = 120576V4

(24)

323 Approximate Symmetries and Optimal System The Case120572 = minus120573 When 120572 = minus120573 the equation admits seven-dimen-sional approximate Lie algebra 1198717 as follows

V1 = 119909 120597120597119909 + 3119905 120597120597119905 minus 2119906 120597

120597119906 + 120576 [31205721199092120597120597119906]

V2 = minus6119905 120597120597119909 + 120597120597119906 + 120576 [minus3120572119905 120597120597119906]

V3 = 120597120597119909 + 120576 [3120572119905 120597

120597119909] V4 = 120597

120597119906V5 = 120576V2V6 = 120576V3V7 = 120576V4

(25)

It is worth nothing that the seven-dimensional approxi-mate Lie algebra 1198717 = 119892 is solvable and its finite sequence ofideals is as follows

0 sub ⟨V6⟩ sub ⟨V5 V6⟩ sub ⟨V3 V5 V6⟩ sub ⟨V2 V3 V5 V6⟩sub ⟨V2 V3 V4 V5 V6⟩ sub ⟨V2 V3 V4 V5 V6 V7⟩ sub 119892 (26)

In the following we will construct the optimal system ofabove Lie algebra 1198717Themethod used here for obtaining theone-dimensional optimal system of subalgebras is that givenin [7] This approach is taking a general element from theLie algebra and reducing it to its simplest equivalent formby applying carefully chosen adjoint transformations that aredefined as follows

Definition 6 (see [7]) Let119866 be a Lie groupAnoptimal systemof s-parameter subgroups is a list of conjugacy inequivalents-parameter subalgebras with the property that any othersubgroup is conjugate to precisely one subgroup in the listSimilarly a list of s-parameter subalgebras forms an optimalsystem if every s-parameter subalgebra of 119892 is equivalent to aunique member of the list under some element of the adjointrepresentation ℎ = 119860119889(119892(ℎ)) 119892 isin 119866Theorem7 (see [7]) Let119867 and be connected s-dimensionalLie subgroups of the Lie group 119866 with corresponding Liesubalgebras ℎ and ℎ of the Lie algebra 119892 of119866Then119867 = 119892119867119892minus1are conjugate subgroups if and only if ℎ = 119860119889(119903(ℎ)) are con-jugate subalgebras

By Theorem 7 the problem of finding an optimal systemof subgroups is equivalent to that of finding an optimal systemof subalgebras To compute the adjoint representation we usethe Lie series

119860119889 (exp (120583V119894)) V119895 = V119895 minus 120583 [V119894 V119895] + 12058322 [V119894 [V119894 V119895]]

minus sdot sdot sdot = exp (119886119889 (minus120583V119894)) V119895(27)

where [V119894 V119895] is the commutator for the Lie algebra and 120583 isa parameterand 119894 119895 = 1 7 In this manner we constructTable 4 with the (119894 119895)th entry indicating 119860119889(exp(120583V119894))V119895Theorem 8 An optimal system of one-dimensional approx-imate symmetry algebra (case 120572 = minus120573) of equation (1) isprovided by V7 V6 V1 + 119887V6 V3 + 119887V7 V2 + 119887V5 + 119888V7 V5 + 119887V3 +119888V7 V4 + 119887V2 + 119888V5 + 119889V7Proof Considering the approximate symmetry algebra g of(1) whose adjoint representation was determined in theTable 4 our task is to simplify as many of the coefficients 119886119894 aspossible through judicious applications of adjoint maps to119881119894so that 119881119894 is equivalent 1198811015840119894 under the adjoint representation

Given a nonzero vector

1198811 = 1198861V1 + 1198862V2 + 1198863V3 + 1198864V4 + 1198865V5 + 1198866V6 + 1198867V7 (28)

Advances in Mathematical Physics 5

Table 1 Approximate commutators of approximate symmetry algebra of (1) (119886 = 2120572 minus 120573 119887 = 2120573 minus 120572 119888 = 120572 + 120573 119889 = 31205722)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus119889V5 minus V3 minus3V4 + 3120576119888V1 2V5 minusV6 minus3V7V2 minus2V2 0 0 6V3 minus 119887V5 0 0 6V6V3 119889V5 + V3 0 0 minus119886V6 0 0 0V4 3V4 minus 3120576119888V1 minus6V3 + 119887V5 119886V6 0 minus6V6 0 0V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 2 Approximate commutators of approximate symmetry algebra of (1) (120572 = 2120573 119886 = 3120572)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus1198862 V5 minus V3 minus3V4 + 3120576119886V1 2V5 minusV6 minus3V7

V2 minus2V2 0 0 minus6V3 + 52 V5 0 0 6V6

V3 V3 + 52V5 0 0 minus119886V6 0 0 0

V4 3V4 minus 3120576119886V1 6V3 minus 52 V5 119886V6 0 minus6V6 0 0

V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 3 Approximate commutators of approximate symmetry algebra of (1) ( 120572 = minus120573 119886 = 3120572)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus1198862 V5 minus V3 minus3V4 2V5 minusV6 minus3V7

V2 minus2V2 0 0 6V3 + 119886V5 0 0 6V6V3

1198862 V5 + V3 0 0 minus119886V6 0 0 0

V4 3V4 minus6V3 minus 119886V5 119886V6 0 minus6V6 0 0V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 4 Adjoint representation of approximate symmetry of (1)

119860119889 V1 V2 V3 V4 V5 V6 V7V1 V1 119890minus2120583V2 119890minus120583V3 + 119886

2 119890minus120583V5 minus1198862 V5 1198903120583V4 119890minus2120583V5 119890minus120583V6 1198903120583V7

V2 V1 + 2120583V2 V2 V3 V4 minus 120583(6V3 + 119886V5) V5 V6 V7 minus 6120583V6V3 V1 minus 120583(119886V52 + V3) V2 V3 V4 + 120583(119886V6) V5 V6 V7V4 V1 minus 3120583V4 V2 + 120583(6V3 + 119886V5) V3 minus 119886120583V6 V4 V5 + 6120583V6 V6 V7V5 V1 + 2120583V5 V2 V3 V4 minus 6120583V6 V5 V6 V7V6 V1 minus 120583V6 V2 V3 V4 V5 V6 V7V7 V1 minus 3120583V7 V2 + 6120583V7 V3 V4 V5 V6 V7

First suppose that 1198861 = 0 Scaling 1198811 if necessary we canassume that 1198861 = 1 As for the 7th column of the Table 4we have

11988110158401 = 119860119889(exp ((11988643 V4))) ∘ 119860119889(exp ((minus11988622 V2)))∘ 119860119889 (exp ((11988673 V7)))

∘ 119860119889 (exp ((1198863 + 211988621198864) V3)))∘ 119860119889 (exp ((minus11988652 + 11988611988634 + 1198861198862119886412 ) V5)))1198811

= V1 + (1198866 minus 119886119886311988643 + 211988641198865 + 311988621198867) V6(29)

6 Advances in Mathematical Physics

The remaining approximate one-dimensional subalgebras arespanned by vectors of the above form with 1198861 = 0 If 1198864 = 0we have

1198812 = 1198862V2 + 1198863V3 + V4 + 1198865V5 + 1198866V6 + 1198867V7 (30)

Next we act on 1198812 to cancel the coefficients of V3 and V6 asfollows

11988110158402 = 119860119889(exp((minus119886361198862 V4)))

∘ 119860119889 (exp((minus119886561198862 +11988611988623361198862 +

11988666 ) V5)))1198812= 1198862V2 + V4 + (1198865 minus 11988611988636 V5 + 1198867V7

(31)

If 1198861 1198864 = 0 and 1198862 = 0 the nonzero vector1198813 = V2 + 1198863V3 + 1198865V5 + 1198866V6 + 1198867V7 (32)

is equivalent to

11988110158403 = 119860119889(exp ((minus11988636 V4)))

∘ 119860119889 (exp((minus11988666 + minus1198861198862336 + 119886611988656 ) V7)))1198813= V2 + (1198865 minus 11988611988636 ) V5 + 1198867V7

(33)

If 1198861 1198862 1198864 = 0 and 1198865 = 0 we scale to make 1198865 = 1 and then

1198814 = 1198863V3 + V5 + 1198866V6 + 1198867V7 (34)

is equivalent to 11988110158404 under the adjoint representation11988110158404 = 119860119889(exp(( 119886661198867 V2)))1198814 = 1198863V3 + V5 + 1198867V7 (35)

If 1198861 1198862 1198864 1198865 = 0 and 1198863 = 0 in the same way as before thenonzero vector

1198815 = V3 + 1198866V6 + 1198867V7 (36)

can be simplified as

11988110158405 = 119860119889(exp(( 119886661198867 V2)))1198812 = V3 + 1198867V7 (37)

If 1198861 1198862 1198863 1198864 1198865 = 0 and 1198867 = 0 we can act

1198816 = 1198866V6 + V7 (38)

by119860119889(exp(((11988666)V2))) to cancel the coefficient of V6 leadingto

11988110158406 = 119860119889 (exp ((11988666 V2)))1198816 = V7 (39)

The last remaining case occurs when 1198861 1198862 1198863 1198864 1198865 1198867 =0 and 1198866 = 0 for which our earlier simplifications wereunnecessary because the only remaining vectors are themultiples of 1198866 on which the adjoint representation actstrivially

4 Approximately Invariant Solutions

In this section we use two different techniques to constructnew approximate solutions of (1) when 120572 = minus12057341 Approximately Invariant Solutions I In the beginning ofthis section we compute an approximately invariant solutionbased on the 119883 = V2 The approximate invariants for 119883 aredetermined by

119883 (119869) = (minus6119905 120597120597119909 + 120597

120597119906 + 120576 (minus3120572119905 120597120597119906)) (1198690 + 1205761198691)= 119900 (120576)

(40)

Equivalently

(minus6119905 120597120597119909 + 120597120597119906) (1198690) = 0

(minus3120572119905 120597120597119906) (1198691) + (minus6119905 120597120597119909 + 120597120597119906) (1198690) = 0

(41)

Thefirst equation has two functionally independent solutions1198690 = 119905 and 1198690 = 1199096119905 + 119906 The simplest solutions of thesecond equation are respectively 1198691 = 0 and 1198691 = minus1205721199092 + 119905Therefore we have two independent invariants to 1199096119905 + 119906 +120576(minus1205721199092 + 119905) and 120593(119905) with respect to119883

Letting 1199096119905 + 119906 + 120576(3120572119905119906) = 120593(119905) we obtain 119906 = 120593(119905) minus1199096119905minus120576(minus1205721199092+ 119905) for the approximately invariant solutionsTherefore we can obtain

119906 (119909 119905) asymp minus 1199096119905 + Γ0 (119905) + 120576 (minus119905 minus 120572119909

2 + Γ1 (119905)+ Γ0 (119905) Γ10158401 (119905)

(42)

Putting (42) into (1) we obtain

Γ0 (119905) = 1198621119879Γ1 (119905) = 11986221199053 + 119905 + 1198622119905

(43)

where 1198621 and 1198622 are arbitrary constants Hence we obtainedthe approximately invariant solution to (1)

119906 (119909 119905) asymp minus 1199096119905 +

1198621119905 + 120576 (1198622119905 minus 1205721199092 ) (44)

In this manner we compute approximate invariants withrespect to the generators of Lie algebra and optimal systemas shown in Table 5

42 Approximately Invariant Solution II Now we apply adifferent technique to find approximately invariant solutionsfor (1) We will begin with one exact solution

119906 = 119908 (120577) = minus12

1cosh2120577 (45)

Advances in Mathematical Physics 7

Table 5 Approximate invariants with respect to the generators of Lie algebra and optimal system

Operator Approximate Invariants

V11199051199093 1199092119906 minus 120576(1205721199093

2 )V2 119905 1199096119905 + 119906 + 120576 (minus1205721199092 + 119905)V3 119905 119906V4 119909 119906V5 119905 1199096119905 + 119906V6 119905 119906V7 119909 119906V1 + 119887V6 119905

1199093 + 120576(119887119909minus4119905) 1199092119906 minus 120576(12057211990932 + 2119887119909

3119906 )V3 + 119887V7 119905 minus 120576(119887119909) 119906V4 + 119887V2 + 119888V5 + 119889V7 1199052

2 + 1199096119887 + 120576 (( 119888119887 minus 119889) 119905) 119887119906 minus 119905 + 120576(31205721199052

2 + 119889119905)V2 + 119887V5 + 119888V7 119905 minus 120576 (119906119888 )

1199096119905 + 119906 + 120576(3120572119905119906 minus 1198881199092

721199053 )V5 + 119887V3 + 119888V7 119905 minus 120576 (119888119909119887 ) 119906

of the unperturbed equation 119906119905 minus 6119906119906119909 + 119906119909119909119909 = 0 Here 120577 =(119909 minus 119905)2 is an invariant of the group with the generator

1198830 = 120597120597119905 +

120597120597119909 (46)

The function 119908 given by (45) is invariant under the operator(46)

Using the generators 1198831 = 120597120597119909 admitted by 119906119905 minus 6119906119906119909 +119906119909119909119909 = 0 we will take the approximate symmetry

119883 = 1198830 + 1205761198831 = 120597120597119905 +

120597120597119909 + 120576 120597

120597119909 (47)

and use it looking for the approximately invariant solution of(1) in the form

119906 = 119908 + 120576V (119905 119909) (48)

Then the invariant equation test119883(119906 minus 119908 minus 120576V)|(48) = 119900(120576)can be written as

1198830 (119906 minus 119908) + 120576 [1198831 (119906 minus 119908) minus 1198830 (V)])119906=119908

= 0 (49)

Note that 1198830 does not contain the differentiation in 119906 there-fore (49) becomes

[1198831 (119906 minus 119908) minus 1198830 (V)]119906=119908

= 0 (50)

whence we obtain the following differential for V(119905 119909)V119905 + V119909 = minus (119908)119909 (51)

Because 119908 = 119908(120577) so (51) can be integrated by the variables

120577 = 119909 minus 1199052

120588 = 119905(52)

Then denoting by1199081015840 the derivative of119908with respect to 120577 wehave

(119908)119905 = minus12 (119908)1015840 (119908)119909 = 1

2 (119908)1015840(53)

and (51) becomes

V120588 = minus12 (119908)1015840 (54)

The integration yields

V = minus12120588 (119908)1015840 + 120595 (119909) (55)

Returning to the variables 119905 119909 we haveV = minus119905 (119908)119909 + 120595 (119909) (56)

Inserting this V in (48) and substituting it into (1) we obtain120595 = ((minus1 minus 3120573)6)119909Thus the approximate symmetry (47) provides the fol-

lowing approximately invariant solution (approximate trav-elling wave)

119906 (119909 119905) = minus12 sdot 1

cosh2 ((119909 minus 119905) 2)+ 120576 [minus 119905

21

cosh2 ((119909 minus 119905) 2) sdot tanh (119909 minus 1199052 )

+ minus1 minus 31205736 119909]

(57)

8 Advances in Mathematical Physics

5 Approximate Conservation Laws

Approximate Lie symmetry can be used to construct theapproximate conservation laws but in this section wewill usepartial Lagrangians to construct approximate conservationlaws of (1) this is a more concise method

Definition 9 (see [10]) An operator 119884 is a kth-order approx-imate Lie-Backlund symmetry

119884 = 1198840 + 1205761198841 + sdot sdot sdot + 120576119896119884119896 (58)

where

119884 = 120585119894 120597120597119909119894 + 120578119894 120597120597119906120572 119894 = 1 2 119899 (59)

and

119884119887 = 120585119894119887 120597120597119909119894 + 120578120572119887 120597

120597119906120572 + 120577120572119887119894 120597120597119906120572119894 + 12057712057211988711989411198942 12059712059711990612057211989411198942 + sdot sdot sdot 119887 = 0 119896

(60)

where 120585119894119887 120578120572119887 isin 119860 and the additional coefficients are

120577120572119887119894 = 119863119894 (119882120572119887 ) + 12058511989511988711990612057211989411989512057712057211988711989411198942 = 11986311989411198631198942 (119882120572119887 ) + 120585119895

11988711990612057211989511989411198942 119894 119895 = 1 119899 (61)

and119882120572119887 is the Lie characteristic function defined by

119882120572119887 = 120578120572119887 minus 120585119895119887119906120572119895 (62)

Definition 10 (see [10]) The approximate Noether operatorassociated with an approximate Lie-Backlund symmetryoperator 119884 is given by

119873119894 = 120585119894 +119882120572 120575120575119906120572 +sum

119904ge1

1198631198941 119863119894s (119882120572) 12057512057511990612057211989411989411198942 119894119904 119894 = 1 119899

(63)

where

119873119894 = 1198731198940 + 1205761198731198941 + sdot sdot sdot + 120576119896119873119894119896 119894 = 1 119899 (64)

where119863119894 is the total derivative operator defined as

119863119894 = 120597120597119909119894 + 119906119894 120597120597119906 + 119906119894119895 120597

120597119906119895 + sdot sdot sdot + 11990611989411989411198942119894119899 12059712059711990611989411198942119894119899

+ sdot sdot sdot 119894 = 1 2 119899(65)

and here119873119894119887 119887 = 0 119896 areNoether operators and the Euler-Lagrange operators are

120575120575119906120572 =

120597120597119906120572 +sum

119904ge1

(minus1)1199041198631198951 119863119895s 12059712059711990612057211989411989511198952119895119904

120572 = 1 119898 119894 = 1 119899(66)

The Euler-Lagrange approximate Lie-Backlund and approx-imate Noether operators are connected by the operatoridentity

119884 + 119863119894 (120585119894) = 119882120572 120575120575119906120572 + 119863119894119873119894 (67)

Definition 11 (see [10]) If there exists a function 119871 = 119871(119909 119906119906(1) 119906(119897) isin 119860 119897 lt 119903 and nonzero functions 119891120573120574 isin 119860 suchthat (1) which can be written as 120575119871120575119906120573 = 120576119891120573120574 1198641205741 where119891120573120574 = 119891120573120574 (119909 119906 119906(1) 119906(119903minus1) 120573 120574 = 1 119898 is an invertiblematrix then provided that 1198641205741 = 0 some 119871 is called a partialLagrangian otherwise it is the standard Lagrangian We termdifferential equations of the form

120575119871120575119906120573 = 120576119891120573120574 1198641205741 (68)

approximate Euler-Lagrange-type equations

Definition 12 (see [5]) An approximate Lie-Backlund sym-metry operator 119884 is called an approximate Noether-typesymmetry operator corresponding to a partial Lagrangian119871 isin119860 if and only if there exists a vector 119861 = (1198611 1198612 119861119899) 119861119894 isin119860 defined by

119861119894 = 1198611198940 + 1205761198611198941 + sdot sdot sdot + 120576119896119861119894119896 (69)

such that

119884 (119871) + 119871119863119894 (120585119894) = 119882120572 120575119871120575119906120572 + 119863119894 (119861119894) + 119900 (120576119896+1) (70)

where 119882 = (11988211198822 119882119898)119882120573 isin 119860 of 119884 is also thecharacteristic of the conservation law 119863119894119879119894 = 119900(120576119896+1) where

119879119894 = 119861119894 minus 119871120585119894 minus119882120573 120597119871120597119906120573119894 + sdot sdot sdot + 119900 (120576119896+1) (71)

of the approximate Euler-Lagrange-type (68)

Because (1) does not have a Lagrange function and if weput a transform 119906 = V119909 then (1) becomes

V119909119905 minus 6V119909V119909119909 + V119909119909119909119909 + 120576120572 (V119909 + 2119909V119909119909)minus 120576120573 (2V119909 + 119909V119909119909) = 0 (72)

In order to write convenient we can get

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 + 120576120572 (119906119909 + 2119909119906119909119909)minus 120576120573 (2119906119909 + 119909119906119909119909) = 0 (73)

Obviously

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 = 0 (74)

and the Lagrange function is

119871 = 12 (119906119909119909)2 + (119906119909)3 minus 1

2119906119909119906119905 (75)

Advances in Mathematical Physics 9

And the approximate 119864119906119897119890119903 minus 119871119886119892119903119886119899119892119890 minus 119905119910119901119890 equation is

120575119871120575119906 = minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909) (76)

So the approximate Noether symmetry operator is 1198840 + 1205981198841for 119871

(1198840 + 1205761198841) 119871 + 119863119894 (1205851198940 + 1205761205851198941) 119871= [(1205780 minus 1205851198950119906119895) + 120576 (1205781 minus 1205851198951119906119895)]sdot [minus120598120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 119863119894 (1198611198940 + 1205761198611198941)

(77)

where

1198840 + 1205761198841 = (12058510 + 12057612058511) 120597120597119905 + (12058520 + 12057612058521) 120597

120597119909 + (1205780+ 1205761205781) 120597

120597119906 + 120577119905 120597120597119906119905 + 120577119909 120597120597119906119909 + 120577119909119909 120597

120597119906119909119909 (78)

120577119905 = 1205780119905 + (1205780119906 minus 12058510119905) 119906119905 minus 12058520119905119906119909 minus 120585101199061199062119905 minus 12058520119906119906119909119906119905+ 12058510119906119905119905 + 12058520119906119909119905 + 120576 [1205781119905 + (1205781119906 minus 12058511119905) 119906119905 minus 12058521119905119906119909minus 120585111199061199062119905 minus 120585211199061199062119909 + 12058511119906119905119905 + 12058521119906119909119905]

(79)

120577119909 = 1205780119909 + (1205780119906 minus 12058520119909) 119906119909 minus 12058510119909119906119905 minus 120585201199061199062119909 minus 12058510119906119906119909119906119905+ 12058510119906119909119905 + 12058520119906119909119909 + 120576 [1205781119909 + (1205781119906 minus 12058521119909) 119906119909 minus 12058511119909119906119905minus 120585211199061199062119909 minus 12058521119906119906119909119906119905 + 12058511119906119909119911119905 + 12058521119906119909119909]

(80)

120577119909119909 = 1205780119909119909 + (21205780119909119906 minus 12058520119909119909) 119906119909 + (1205780119906119906 minus 212058520119909119906) 1199062119909minus 12058510119909119909119906119905 minus 120585101199061199061199062119909119906119905 minus 212058510119909119906119906119909119906119905 minus 12058510119906119906119909119909119906119905minus 212058510119909119906119909119905 minus 212058510119906119906119909119906119909119905 minus 120585201199061199061199063119909 minus 312058520119906119906119909119906119909119909+ (1205780119906 minus 212058520119909) 119906119909119909 + 12058510119906119909119909119905 + 12058520119906119909119909119909 + 120576 [1205781119909119909+ (21205781119909119906 minus 12058521119909119909) 119906119909 + (1205781119906119906 minus 212058521119909119906) 1199062119909 minus 12058511119909119909119906119905minus 120585111199061199061199062119909119906119905 minus 212058511119909119906119906119909119906119905 minus 12058511119906119906119909119909119906119905 minus 212058511119909119906119909119905minus 212058511119906119906119909119906119909119905 minus 120585211199061199061199063119909 minus 312058521119906119906119909119906119909119909+ (1205781119906 minus 212058521119909) 119906119909119909 + 12058511119906119909119909119905 + 12058521119906119909119909119909]

(81)

So (58) becomes

minus 12119906119909120577119905 minus

12119906119905120577119909 + 31199062119909120577119909 + 119906119909119909120577119909119909

+ 119871 [12058510119905 + 12058510119906119906119905 + 120576 (12058511119905 + 12058511119906119906119905)]+ 119871 [12058520119905 + 12058520119906119906119905 + 120576 (12058521119905 + 12058521119906119906119905)]= [1205780 minus 12058510119906119905 minus 12058520119906119909 + 120576 (1205781 minus 12058511119906119905 minus 12058521119906119909)]

sdot [minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 120576 (11986111119905 + 11986111119906119906119905) + 120576 (11986121119905 + 11986121119906119906119909)+ (11986110119905 + 11986110119906119906119905) + (11986120119909 + 11986120119906119906119909)

(82)

Put 119871 120577119905 120577119909 120577119909119909 into (82) and let 1205762 = 0 We obtain

minus121205780119905 minus 11986120119906 = 01205780119909 = 01205780119906 = 012058510 = 012058520 = 0

11986110119906 = 011986120119909 + 11986110119905 = 0

minus121205781119909 minus 11986111119906 = 01205781119909 = 01205781119906 = 012058511 = 012058521 = 0

11986110119906 = 011986121119909 + 11986111119905 = 0

minus121205781119905 + (120572 minus 2120573) 1205780 minus 11986121119906 = 01205781119909119909 + (2120572 minus 120573) 1205780 sdot 119909 = 0

(83)

In the following we will consider three cases of 120572 and 120573First Case 120572 = 2120573 and 120573 = 2120572

12058510 = 012058520 = 01205780 = 012058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = 119891 (119905 119909)

10 Advances in Mathematical Physics

11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905) 1199062 + 119895 (119905 119909)

(84)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909)119891(119905 119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 and 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetric operators

119884 = 120576119892 (119905) 120597120597119906 (85)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12120576119906119909119892 (119905)

1198792 = 119891 (119905 119909) + 120576 [minus12119892119905 (119905) 119906 + 119895 (119905 119909)]minus [120576119892 (119905)] [31199062119909 minus 1

2119906119905] + 119863119909 [120576119892 (119905)] 119906119909119909(86)

Second Case 120572 = 2120573 and 120573 = 212057212058510 = 012058520 = 01205780 = ℎ (119905) 12058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = minusℎ119905 (119905) 1199062 + 119891 (119905 119909) 11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905)2 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)

(87)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909) 119891(119905119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetry operators

119884 = ℎ (119905) 120597120597119906 + 120576119892 (119905) 120597

120597119906 (88)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12119906119909 [ℎ (119905) + 120576119892 (119905)]

1198792 = 119891 (119905 119909) + [minus12ℎ119905 (119905) 119906]+ 120576 [minus12119892119905 (119905) 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)]minus [120576119892 (119905) + ℎ (119905)] [31199062119909 minus 1

2119906119905]+ 119863119909 [120576119892 (119905) + ℎ (119905)] 119906119909119909

(89)

Third Case 120572 = 2120573 and 120573 = 2120572 the conservation laws are thesame as the first case

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Thisworkwas supported in part by theNatural Science Foun-dation of Inner Mongolia of China under grant 2016MS0116

References

[1] A Wazwaz Partial Differential Equations and Solitary WavesTheory Higher Education Press Beijing China 2009

[2] Y S Kivshar and B A Malomed ldquoDynamics of solitons innearly integrable systemsrdquo Reviews of Modern Physics vol 61no 4 pp 763ndash915 1989

[3] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMatematicheskii Sbornik vol 64 no 2 p 427 1989

[4] V A Bailov R K Gazizov and N H Ibragmov ldquoPertubationmethods in group analysisrdquo Journal of Mathematical Sciencesvol 55 no 1 pp 1450ndash1485 1991

[5] N H Ibragimov and V F Kovalev Approximate and Renor-mgroup Symmetries Higher Education Press Beijing China2009

[6] V A Baikov and N H Ibragimov ldquoContinuation of approxi-mate transformation groups via multiple time scales methodrdquoNonlinear Dynamics vol 22 no 1 pp 3ndash13 2000

[7] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations vol 168 ofApplied Mathematical Sciences Springer New York NY USA2010

[9] A H Kara F M Mahomed and G Unal ldquoApproximate sym-metries and conservation laws with applicationsrdquo InternationalJournal ofTheoretical Physics vol 38 no 9 pp 2389ndash2399 1999

[10] A G Johnpillai A H Kara and FMMahomed ldquoApproximateNoether-type symmetries and conservation laws via partial

Advances in Mathematical Physics 11

Lagrangians for PDEs with a small parameterrdquo Journal of Com-putational and AppliedMathematics vol 223 no 1 pp 508ndash5182009

[11] J-R Su S-L Zhang and J-N Li ldquoApproximate noether-typesymmetries and conservation laws via partial lagrangians fornonlinear wave equation with dampingrdquo Communications inTheoretical Physics vol 53 no 1 pp 37ndash42 2010

[12] E S Benilov and B A Malomed ldquoOn the spectral singularityproblem in the perturbation theory for systems close to theKdVequationrdquo Journal of Physics A Mathematical and General vol21 no 8 pp L439ndashL442 1988

[13] M A Allen and G Rowlands ldquoA solitary-wave solution to aperturbedKdV equationrdquo Journal of Plasma Physics vol 64 no4 pp 475ndash480 2000

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Advances in Mathematical Physics 5

Table 1 Approximate commutators of approximate symmetry algebra of (1) (119886 = 2120572 minus 120573 119887 = 2120573 minus 120572 119888 = 120572 + 120573 119889 = 31205722)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus119889V5 minus V3 minus3V4 + 3120576119888V1 2V5 minusV6 minus3V7V2 minus2V2 0 0 6V3 minus 119887V5 0 0 6V6V3 119889V5 + V3 0 0 minus119886V6 0 0 0V4 3V4 minus 3120576119888V1 minus6V3 + 119887V5 119886V6 0 minus6V6 0 0V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 2 Approximate commutators of approximate symmetry algebra of (1) (120572 = 2120573 119886 = 3120572)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus1198862 V5 minus V3 minus3V4 + 3120576119886V1 2V5 minusV6 minus3V7

V2 minus2V2 0 0 minus6V3 + 52 V5 0 0 6V6

V3 V3 + 52V5 0 0 minus119886V6 0 0 0

V4 3V4 minus 3120576119886V1 6V3 minus 52 V5 119886V6 0 minus6V6 0 0

V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 3 Approximate commutators of approximate symmetry algebra of (1) ( 120572 = minus120573 119886 = 3120572)V1 V2 V3 V4 V5 V6 V7

V1 0 2V2 minus1198862 V5 minus V3 minus3V4 2V5 minusV6 minus3V7

V2 minus2V2 0 0 6V3 + 119886V5 0 0 6V6V3

1198862 V5 + V3 0 0 minus119886V6 0 0 0

V4 3V4 minus6V3 minus 119886V5 119886V6 0 minus6V6 0 0V5 minus2V5 0 0 6V6 0 0 0V6 V6 0 0 0 0 0 0V7 3V7 minus6V6 0 0 0 0 0

Table 4 Adjoint representation of approximate symmetry of (1)

119860119889 V1 V2 V3 V4 V5 V6 V7V1 V1 119890minus2120583V2 119890minus120583V3 + 119886

2 119890minus120583V5 minus1198862 V5 1198903120583V4 119890minus2120583V5 119890minus120583V6 1198903120583V7

V2 V1 + 2120583V2 V2 V3 V4 minus 120583(6V3 + 119886V5) V5 V6 V7 minus 6120583V6V3 V1 minus 120583(119886V52 + V3) V2 V3 V4 + 120583(119886V6) V5 V6 V7V4 V1 minus 3120583V4 V2 + 120583(6V3 + 119886V5) V3 minus 119886120583V6 V4 V5 + 6120583V6 V6 V7V5 V1 + 2120583V5 V2 V3 V4 minus 6120583V6 V5 V6 V7V6 V1 minus 120583V6 V2 V3 V4 V5 V6 V7V7 V1 minus 3120583V7 V2 + 6120583V7 V3 V4 V5 V6 V7

First suppose that 1198861 = 0 Scaling 1198811 if necessary we canassume that 1198861 = 1 As for the 7th column of the Table 4we have

11988110158401 = 119860119889(exp ((11988643 V4))) ∘ 119860119889(exp ((minus11988622 V2)))∘ 119860119889 (exp ((11988673 V7)))

∘ 119860119889 (exp ((1198863 + 211988621198864) V3)))∘ 119860119889 (exp ((minus11988652 + 11988611988634 + 1198861198862119886412 ) V5)))1198811

= V1 + (1198866 minus 119886119886311988643 + 211988641198865 + 311988621198867) V6(29)

6 Advances in Mathematical Physics

The remaining approximate one-dimensional subalgebras arespanned by vectors of the above form with 1198861 = 0 If 1198864 = 0we have

1198812 = 1198862V2 + 1198863V3 + V4 + 1198865V5 + 1198866V6 + 1198867V7 (30)

Next we act on 1198812 to cancel the coefficients of V3 and V6 asfollows

11988110158402 = 119860119889(exp((minus119886361198862 V4)))

∘ 119860119889 (exp((minus119886561198862 +11988611988623361198862 +

11988666 ) V5)))1198812= 1198862V2 + V4 + (1198865 minus 11988611988636 V5 + 1198867V7

(31)

If 1198861 1198864 = 0 and 1198862 = 0 the nonzero vector1198813 = V2 + 1198863V3 + 1198865V5 + 1198866V6 + 1198867V7 (32)

is equivalent to

11988110158403 = 119860119889(exp ((minus11988636 V4)))

∘ 119860119889 (exp((minus11988666 + minus1198861198862336 + 119886611988656 ) V7)))1198813= V2 + (1198865 minus 11988611988636 ) V5 + 1198867V7

(33)

If 1198861 1198862 1198864 = 0 and 1198865 = 0 we scale to make 1198865 = 1 and then

1198814 = 1198863V3 + V5 + 1198866V6 + 1198867V7 (34)

is equivalent to 11988110158404 under the adjoint representation11988110158404 = 119860119889(exp(( 119886661198867 V2)))1198814 = 1198863V3 + V5 + 1198867V7 (35)

If 1198861 1198862 1198864 1198865 = 0 and 1198863 = 0 in the same way as before thenonzero vector

1198815 = V3 + 1198866V6 + 1198867V7 (36)

can be simplified as

11988110158405 = 119860119889(exp(( 119886661198867 V2)))1198812 = V3 + 1198867V7 (37)

If 1198861 1198862 1198863 1198864 1198865 = 0 and 1198867 = 0 we can act

1198816 = 1198866V6 + V7 (38)

by119860119889(exp(((11988666)V2))) to cancel the coefficient of V6 leadingto

11988110158406 = 119860119889 (exp ((11988666 V2)))1198816 = V7 (39)

The last remaining case occurs when 1198861 1198862 1198863 1198864 1198865 1198867 =0 and 1198866 = 0 for which our earlier simplifications wereunnecessary because the only remaining vectors are themultiples of 1198866 on which the adjoint representation actstrivially

4 Approximately Invariant Solutions

In this section we use two different techniques to constructnew approximate solutions of (1) when 120572 = minus12057341 Approximately Invariant Solutions I In the beginning ofthis section we compute an approximately invariant solutionbased on the 119883 = V2 The approximate invariants for 119883 aredetermined by

119883 (119869) = (minus6119905 120597120597119909 + 120597

120597119906 + 120576 (minus3120572119905 120597120597119906)) (1198690 + 1205761198691)= 119900 (120576)

(40)

Equivalently

(minus6119905 120597120597119909 + 120597120597119906) (1198690) = 0

(minus3120572119905 120597120597119906) (1198691) + (minus6119905 120597120597119909 + 120597120597119906) (1198690) = 0

(41)

Thefirst equation has two functionally independent solutions1198690 = 119905 and 1198690 = 1199096119905 + 119906 The simplest solutions of thesecond equation are respectively 1198691 = 0 and 1198691 = minus1205721199092 + 119905Therefore we have two independent invariants to 1199096119905 + 119906 +120576(minus1205721199092 + 119905) and 120593(119905) with respect to119883

Letting 1199096119905 + 119906 + 120576(3120572119905119906) = 120593(119905) we obtain 119906 = 120593(119905) minus1199096119905minus120576(minus1205721199092+ 119905) for the approximately invariant solutionsTherefore we can obtain

119906 (119909 119905) asymp minus 1199096119905 + Γ0 (119905) + 120576 (minus119905 minus 120572119909

2 + Γ1 (119905)+ Γ0 (119905) Γ10158401 (119905)

(42)

Putting (42) into (1) we obtain

Γ0 (119905) = 1198621119879Γ1 (119905) = 11986221199053 + 119905 + 1198622119905

(43)

where 1198621 and 1198622 are arbitrary constants Hence we obtainedthe approximately invariant solution to (1)

119906 (119909 119905) asymp minus 1199096119905 +

1198621119905 + 120576 (1198622119905 minus 1205721199092 ) (44)

In this manner we compute approximate invariants withrespect to the generators of Lie algebra and optimal systemas shown in Table 5

42 Approximately Invariant Solution II Now we apply adifferent technique to find approximately invariant solutionsfor (1) We will begin with one exact solution

119906 = 119908 (120577) = minus12

1cosh2120577 (45)

Advances in Mathematical Physics 7

Table 5 Approximate invariants with respect to the generators of Lie algebra and optimal system

Operator Approximate Invariants

V11199051199093 1199092119906 minus 120576(1205721199093

2 )V2 119905 1199096119905 + 119906 + 120576 (minus1205721199092 + 119905)V3 119905 119906V4 119909 119906V5 119905 1199096119905 + 119906V6 119905 119906V7 119909 119906V1 + 119887V6 119905

1199093 + 120576(119887119909minus4119905) 1199092119906 minus 120576(12057211990932 + 2119887119909

3119906 )V3 + 119887V7 119905 minus 120576(119887119909) 119906V4 + 119887V2 + 119888V5 + 119889V7 1199052

2 + 1199096119887 + 120576 (( 119888119887 minus 119889) 119905) 119887119906 minus 119905 + 120576(31205721199052

2 + 119889119905)V2 + 119887V5 + 119888V7 119905 minus 120576 (119906119888 )

1199096119905 + 119906 + 120576(3120572119905119906 minus 1198881199092

721199053 )V5 + 119887V3 + 119888V7 119905 minus 120576 (119888119909119887 ) 119906

of the unperturbed equation 119906119905 minus 6119906119906119909 + 119906119909119909119909 = 0 Here 120577 =(119909 minus 119905)2 is an invariant of the group with the generator

1198830 = 120597120597119905 +

120597120597119909 (46)

The function 119908 given by (45) is invariant under the operator(46)

Using the generators 1198831 = 120597120597119909 admitted by 119906119905 minus 6119906119906119909 +119906119909119909119909 = 0 we will take the approximate symmetry

119883 = 1198830 + 1205761198831 = 120597120597119905 +

120597120597119909 + 120576 120597

120597119909 (47)

and use it looking for the approximately invariant solution of(1) in the form

119906 = 119908 + 120576V (119905 119909) (48)

Then the invariant equation test119883(119906 minus 119908 minus 120576V)|(48) = 119900(120576)can be written as

1198830 (119906 minus 119908) + 120576 [1198831 (119906 minus 119908) minus 1198830 (V)])119906=119908

= 0 (49)

Note that 1198830 does not contain the differentiation in 119906 there-fore (49) becomes

[1198831 (119906 minus 119908) minus 1198830 (V)]119906=119908

= 0 (50)

whence we obtain the following differential for V(119905 119909)V119905 + V119909 = minus (119908)119909 (51)

Because 119908 = 119908(120577) so (51) can be integrated by the variables

120577 = 119909 minus 1199052

120588 = 119905(52)

Then denoting by1199081015840 the derivative of119908with respect to 120577 wehave

(119908)119905 = minus12 (119908)1015840 (119908)119909 = 1

2 (119908)1015840(53)

and (51) becomes

V120588 = minus12 (119908)1015840 (54)

The integration yields

V = minus12120588 (119908)1015840 + 120595 (119909) (55)

Returning to the variables 119905 119909 we haveV = minus119905 (119908)119909 + 120595 (119909) (56)

Inserting this V in (48) and substituting it into (1) we obtain120595 = ((minus1 minus 3120573)6)119909Thus the approximate symmetry (47) provides the fol-

lowing approximately invariant solution (approximate trav-elling wave)

119906 (119909 119905) = minus12 sdot 1

cosh2 ((119909 minus 119905) 2)+ 120576 [minus 119905

21

cosh2 ((119909 minus 119905) 2) sdot tanh (119909 minus 1199052 )

+ minus1 minus 31205736 119909]

(57)

8 Advances in Mathematical Physics

5 Approximate Conservation Laws

Approximate Lie symmetry can be used to construct theapproximate conservation laws but in this section wewill usepartial Lagrangians to construct approximate conservationlaws of (1) this is a more concise method

Definition 9 (see [10]) An operator 119884 is a kth-order approx-imate Lie-Backlund symmetry

119884 = 1198840 + 1205761198841 + sdot sdot sdot + 120576119896119884119896 (58)

where

119884 = 120585119894 120597120597119909119894 + 120578119894 120597120597119906120572 119894 = 1 2 119899 (59)

and

119884119887 = 120585119894119887 120597120597119909119894 + 120578120572119887 120597

120597119906120572 + 120577120572119887119894 120597120597119906120572119894 + 12057712057211988711989411198942 12059712059711990612057211989411198942 + sdot sdot sdot 119887 = 0 119896

(60)

where 120585119894119887 120578120572119887 isin 119860 and the additional coefficients are

120577120572119887119894 = 119863119894 (119882120572119887 ) + 12058511989511988711990612057211989411989512057712057211988711989411198942 = 11986311989411198631198942 (119882120572119887 ) + 120585119895

11988711990612057211989511989411198942 119894 119895 = 1 119899 (61)

and119882120572119887 is the Lie characteristic function defined by

119882120572119887 = 120578120572119887 minus 120585119895119887119906120572119895 (62)

Definition 10 (see [10]) The approximate Noether operatorassociated with an approximate Lie-Backlund symmetryoperator 119884 is given by

119873119894 = 120585119894 +119882120572 120575120575119906120572 +sum

119904ge1

1198631198941 119863119894s (119882120572) 12057512057511990612057211989411989411198942 119894119904 119894 = 1 119899

(63)

where

119873119894 = 1198731198940 + 1205761198731198941 + sdot sdot sdot + 120576119896119873119894119896 119894 = 1 119899 (64)

where119863119894 is the total derivative operator defined as

119863119894 = 120597120597119909119894 + 119906119894 120597120597119906 + 119906119894119895 120597

120597119906119895 + sdot sdot sdot + 11990611989411989411198942119894119899 12059712059711990611989411198942119894119899

+ sdot sdot sdot 119894 = 1 2 119899(65)

and here119873119894119887 119887 = 0 119896 areNoether operators and the Euler-Lagrange operators are

120575120575119906120572 =

120597120597119906120572 +sum

119904ge1

(minus1)1199041198631198951 119863119895s 12059712059711990612057211989411989511198952119895119904

120572 = 1 119898 119894 = 1 119899(66)

The Euler-Lagrange approximate Lie-Backlund and approx-imate Noether operators are connected by the operatoridentity

119884 + 119863119894 (120585119894) = 119882120572 120575120575119906120572 + 119863119894119873119894 (67)

Definition 11 (see [10]) If there exists a function 119871 = 119871(119909 119906119906(1) 119906(119897) isin 119860 119897 lt 119903 and nonzero functions 119891120573120574 isin 119860 suchthat (1) which can be written as 120575119871120575119906120573 = 120576119891120573120574 1198641205741 where119891120573120574 = 119891120573120574 (119909 119906 119906(1) 119906(119903minus1) 120573 120574 = 1 119898 is an invertiblematrix then provided that 1198641205741 = 0 some 119871 is called a partialLagrangian otherwise it is the standard Lagrangian We termdifferential equations of the form

120575119871120575119906120573 = 120576119891120573120574 1198641205741 (68)

approximate Euler-Lagrange-type equations

Definition 12 (see [5]) An approximate Lie-Backlund sym-metry operator 119884 is called an approximate Noether-typesymmetry operator corresponding to a partial Lagrangian119871 isin119860 if and only if there exists a vector 119861 = (1198611 1198612 119861119899) 119861119894 isin119860 defined by

119861119894 = 1198611198940 + 1205761198611198941 + sdot sdot sdot + 120576119896119861119894119896 (69)

such that

119884 (119871) + 119871119863119894 (120585119894) = 119882120572 120575119871120575119906120572 + 119863119894 (119861119894) + 119900 (120576119896+1) (70)

where 119882 = (11988211198822 119882119898)119882120573 isin 119860 of 119884 is also thecharacteristic of the conservation law 119863119894119879119894 = 119900(120576119896+1) where

119879119894 = 119861119894 minus 119871120585119894 minus119882120573 120597119871120597119906120573119894 + sdot sdot sdot + 119900 (120576119896+1) (71)

of the approximate Euler-Lagrange-type (68)

Because (1) does not have a Lagrange function and if weput a transform 119906 = V119909 then (1) becomes

V119909119905 minus 6V119909V119909119909 + V119909119909119909119909 + 120576120572 (V119909 + 2119909V119909119909)minus 120576120573 (2V119909 + 119909V119909119909) = 0 (72)

In order to write convenient we can get

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 + 120576120572 (119906119909 + 2119909119906119909119909)minus 120576120573 (2119906119909 + 119909119906119909119909) = 0 (73)

Obviously

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 = 0 (74)

and the Lagrange function is

119871 = 12 (119906119909119909)2 + (119906119909)3 minus 1

2119906119909119906119905 (75)

Advances in Mathematical Physics 9

And the approximate 119864119906119897119890119903 minus 119871119886119892119903119886119899119892119890 minus 119905119910119901119890 equation is

120575119871120575119906 = minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909) (76)

So the approximate Noether symmetry operator is 1198840 + 1205981198841for 119871

(1198840 + 1205761198841) 119871 + 119863119894 (1205851198940 + 1205761205851198941) 119871= [(1205780 minus 1205851198950119906119895) + 120576 (1205781 minus 1205851198951119906119895)]sdot [minus120598120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 119863119894 (1198611198940 + 1205761198611198941)

(77)

where

1198840 + 1205761198841 = (12058510 + 12057612058511) 120597120597119905 + (12058520 + 12057612058521) 120597

120597119909 + (1205780+ 1205761205781) 120597

120597119906 + 120577119905 120597120597119906119905 + 120577119909 120597120597119906119909 + 120577119909119909 120597

120597119906119909119909 (78)

120577119905 = 1205780119905 + (1205780119906 minus 12058510119905) 119906119905 minus 12058520119905119906119909 minus 120585101199061199062119905 minus 12058520119906119906119909119906119905+ 12058510119906119905119905 + 12058520119906119909119905 + 120576 [1205781119905 + (1205781119906 minus 12058511119905) 119906119905 minus 12058521119905119906119909minus 120585111199061199062119905 minus 120585211199061199062119909 + 12058511119906119905119905 + 12058521119906119909119905]

(79)

120577119909 = 1205780119909 + (1205780119906 minus 12058520119909) 119906119909 minus 12058510119909119906119905 minus 120585201199061199062119909 minus 12058510119906119906119909119906119905+ 12058510119906119909119905 + 12058520119906119909119909 + 120576 [1205781119909 + (1205781119906 minus 12058521119909) 119906119909 minus 12058511119909119906119905minus 120585211199061199062119909 minus 12058521119906119906119909119906119905 + 12058511119906119909119911119905 + 12058521119906119909119909]

(80)

120577119909119909 = 1205780119909119909 + (21205780119909119906 minus 12058520119909119909) 119906119909 + (1205780119906119906 minus 212058520119909119906) 1199062119909minus 12058510119909119909119906119905 minus 120585101199061199061199062119909119906119905 minus 212058510119909119906119906119909119906119905 minus 12058510119906119906119909119909119906119905minus 212058510119909119906119909119905 minus 212058510119906119906119909119906119909119905 minus 120585201199061199061199063119909 minus 312058520119906119906119909119906119909119909+ (1205780119906 minus 212058520119909) 119906119909119909 + 12058510119906119909119909119905 + 12058520119906119909119909119909 + 120576 [1205781119909119909+ (21205781119909119906 minus 12058521119909119909) 119906119909 + (1205781119906119906 minus 212058521119909119906) 1199062119909 minus 12058511119909119909119906119905minus 120585111199061199061199062119909119906119905 minus 212058511119909119906119906119909119906119905 minus 12058511119906119906119909119909119906119905 minus 212058511119909119906119909119905minus 212058511119906119906119909119906119909119905 minus 120585211199061199061199063119909 minus 312058521119906119906119909119906119909119909+ (1205781119906 minus 212058521119909) 119906119909119909 + 12058511119906119909119909119905 + 12058521119906119909119909119909]

(81)

So (58) becomes

minus 12119906119909120577119905 minus

12119906119905120577119909 + 31199062119909120577119909 + 119906119909119909120577119909119909

+ 119871 [12058510119905 + 12058510119906119906119905 + 120576 (12058511119905 + 12058511119906119906119905)]+ 119871 [12058520119905 + 12058520119906119906119905 + 120576 (12058521119905 + 12058521119906119906119905)]= [1205780 minus 12058510119906119905 minus 12058520119906119909 + 120576 (1205781 minus 12058511119906119905 minus 12058521119906119909)]

sdot [minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 120576 (11986111119905 + 11986111119906119906119905) + 120576 (11986121119905 + 11986121119906119906119909)+ (11986110119905 + 11986110119906119906119905) + (11986120119909 + 11986120119906119906119909)

(82)

Put 119871 120577119905 120577119909 120577119909119909 into (82) and let 1205762 = 0 We obtain

minus121205780119905 minus 11986120119906 = 01205780119909 = 01205780119906 = 012058510 = 012058520 = 0

11986110119906 = 011986120119909 + 11986110119905 = 0

minus121205781119909 minus 11986111119906 = 01205781119909 = 01205781119906 = 012058511 = 012058521 = 0

11986110119906 = 011986121119909 + 11986111119905 = 0

minus121205781119905 + (120572 minus 2120573) 1205780 minus 11986121119906 = 01205781119909119909 + (2120572 minus 120573) 1205780 sdot 119909 = 0

(83)

In the following we will consider three cases of 120572 and 120573First Case 120572 = 2120573 and 120573 = 2120572

12058510 = 012058520 = 01205780 = 012058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = 119891 (119905 119909)

10 Advances in Mathematical Physics

11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905) 1199062 + 119895 (119905 119909)

(84)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909)119891(119905 119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 and 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetric operators

119884 = 120576119892 (119905) 120597120597119906 (85)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12120576119906119909119892 (119905)

1198792 = 119891 (119905 119909) + 120576 [minus12119892119905 (119905) 119906 + 119895 (119905 119909)]minus [120576119892 (119905)] [31199062119909 minus 1

2119906119905] + 119863119909 [120576119892 (119905)] 119906119909119909(86)

Second Case 120572 = 2120573 and 120573 = 212057212058510 = 012058520 = 01205780 = ℎ (119905) 12058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = minusℎ119905 (119905) 1199062 + 119891 (119905 119909) 11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905)2 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)

(87)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909) 119891(119905119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetry operators

119884 = ℎ (119905) 120597120597119906 + 120576119892 (119905) 120597

120597119906 (88)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12119906119909 [ℎ (119905) + 120576119892 (119905)]

1198792 = 119891 (119905 119909) + [minus12ℎ119905 (119905) 119906]+ 120576 [minus12119892119905 (119905) 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)]minus [120576119892 (119905) + ℎ (119905)] [31199062119909 minus 1

2119906119905]+ 119863119909 [120576119892 (119905) + ℎ (119905)] 119906119909119909

(89)

Third Case 120572 = 2120573 and 120573 = 2120572 the conservation laws are thesame as the first case

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Thisworkwas supported in part by theNatural Science Foun-dation of Inner Mongolia of China under grant 2016MS0116

References

[1] A Wazwaz Partial Differential Equations and Solitary WavesTheory Higher Education Press Beijing China 2009

[2] Y S Kivshar and B A Malomed ldquoDynamics of solitons innearly integrable systemsrdquo Reviews of Modern Physics vol 61no 4 pp 763ndash915 1989

[3] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMatematicheskii Sbornik vol 64 no 2 p 427 1989

[4] V A Bailov R K Gazizov and N H Ibragmov ldquoPertubationmethods in group analysisrdquo Journal of Mathematical Sciencesvol 55 no 1 pp 1450ndash1485 1991

[5] N H Ibragimov and V F Kovalev Approximate and Renor-mgroup Symmetries Higher Education Press Beijing China2009

[6] V A Baikov and N H Ibragimov ldquoContinuation of approxi-mate transformation groups via multiple time scales methodrdquoNonlinear Dynamics vol 22 no 1 pp 3ndash13 2000

[7] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations vol 168 ofApplied Mathematical Sciences Springer New York NY USA2010

[9] A H Kara F M Mahomed and G Unal ldquoApproximate sym-metries and conservation laws with applicationsrdquo InternationalJournal ofTheoretical Physics vol 38 no 9 pp 2389ndash2399 1999

[10] A G Johnpillai A H Kara and FMMahomed ldquoApproximateNoether-type symmetries and conservation laws via partial

Advances in Mathematical Physics 11

Lagrangians for PDEs with a small parameterrdquo Journal of Com-putational and AppliedMathematics vol 223 no 1 pp 508ndash5182009

[11] J-R Su S-L Zhang and J-N Li ldquoApproximate noether-typesymmetries and conservation laws via partial lagrangians fornonlinear wave equation with dampingrdquo Communications inTheoretical Physics vol 53 no 1 pp 37ndash42 2010

[12] E S Benilov and B A Malomed ldquoOn the spectral singularityproblem in the perturbation theory for systems close to theKdVequationrdquo Journal of Physics A Mathematical and General vol21 no 8 pp L439ndashL442 1988

[13] M A Allen and G Rowlands ldquoA solitary-wave solution to aperturbedKdV equationrdquo Journal of Plasma Physics vol 64 no4 pp 475ndash480 2000

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6 Advances in Mathematical Physics

The remaining approximate one-dimensional subalgebras arespanned by vectors of the above form with 1198861 = 0 If 1198864 = 0we have

1198812 = 1198862V2 + 1198863V3 + V4 + 1198865V5 + 1198866V6 + 1198867V7 (30)

Next we act on 1198812 to cancel the coefficients of V3 and V6 asfollows

11988110158402 = 119860119889(exp((minus119886361198862 V4)))

∘ 119860119889 (exp((minus119886561198862 +11988611988623361198862 +

11988666 ) V5)))1198812= 1198862V2 + V4 + (1198865 minus 11988611988636 V5 + 1198867V7

(31)

If 1198861 1198864 = 0 and 1198862 = 0 the nonzero vector1198813 = V2 + 1198863V3 + 1198865V5 + 1198866V6 + 1198867V7 (32)

is equivalent to

11988110158403 = 119860119889(exp ((minus11988636 V4)))

∘ 119860119889 (exp((minus11988666 + minus1198861198862336 + 119886611988656 ) V7)))1198813= V2 + (1198865 minus 11988611988636 ) V5 + 1198867V7

(33)

If 1198861 1198862 1198864 = 0 and 1198865 = 0 we scale to make 1198865 = 1 and then

1198814 = 1198863V3 + V5 + 1198866V6 + 1198867V7 (34)

is equivalent to 11988110158404 under the adjoint representation11988110158404 = 119860119889(exp(( 119886661198867 V2)))1198814 = 1198863V3 + V5 + 1198867V7 (35)

If 1198861 1198862 1198864 1198865 = 0 and 1198863 = 0 in the same way as before thenonzero vector

1198815 = V3 + 1198866V6 + 1198867V7 (36)

can be simplified as

11988110158405 = 119860119889(exp(( 119886661198867 V2)))1198812 = V3 + 1198867V7 (37)

If 1198861 1198862 1198863 1198864 1198865 = 0 and 1198867 = 0 we can act

1198816 = 1198866V6 + V7 (38)

by119860119889(exp(((11988666)V2))) to cancel the coefficient of V6 leadingto

11988110158406 = 119860119889 (exp ((11988666 V2)))1198816 = V7 (39)

The last remaining case occurs when 1198861 1198862 1198863 1198864 1198865 1198867 =0 and 1198866 = 0 for which our earlier simplifications wereunnecessary because the only remaining vectors are themultiples of 1198866 on which the adjoint representation actstrivially

4 Approximately Invariant Solutions

In this section we use two different techniques to constructnew approximate solutions of (1) when 120572 = minus12057341 Approximately Invariant Solutions I In the beginning ofthis section we compute an approximately invariant solutionbased on the 119883 = V2 The approximate invariants for 119883 aredetermined by

119883 (119869) = (minus6119905 120597120597119909 + 120597

120597119906 + 120576 (minus3120572119905 120597120597119906)) (1198690 + 1205761198691)= 119900 (120576)

(40)

Equivalently

(minus6119905 120597120597119909 + 120597120597119906) (1198690) = 0

(minus3120572119905 120597120597119906) (1198691) + (minus6119905 120597120597119909 + 120597120597119906) (1198690) = 0

(41)

Thefirst equation has two functionally independent solutions1198690 = 119905 and 1198690 = 1199096119905 + 119906 The simplest solutions of thesecond equation are respectively 1198691 = 0 and 1198691 = minus1205721199092 + 119905Therefore we have two independent invariants to 1199096119905 + 119906 +120576(minus1205721199092 + 119905) and 120593(119905) with respect to119883

Letting 1199096119905 + 119906 + 120576(3120572119905119906) = 120593(119905) we obtain 119906 = 120593(119905) minus1199096119905minus120576(minus1205721199092+ 119905) for the approximately invariant solutionsTherefore we can obtain

119906 (119909 119905) asymp minus 1199096119905 + Γ0 (119905) + 120576 (minus119905 minus 120572119909

2 + Γ1 (119905)+ Γ0 (119905) Γ10158401 (119905)

(42)

Putting (42) into (1) we obtain

Γ0 (119905) = 1198621119879Γ1 (119905) = 11986221199053 + 119905 + 1198622119905

(43)

where 1198621 and 1198622 are arbitrary constants Hence we obtainedthe approximately invariant solution to (1)

119906 (119909 119905) asymp minus 1199096119905 +

1198621119905 + 120576 (1198622119905 minus 1205721199092 ) (44)

In this manner we compute approximate invariants withrespect to the generators of Lie algebra and optimal systemas shown in Table 5

42 Approximately Invariant Solution II Now we apply adifferent technique to find approximately invariant solutionsfor (1) We will begin with one exact solution

119906 = 119908 (120577) = minus12

1cosh2120577 (45)

Advances in Mathematical Physics 7

Table 5 Approximate invariants with respect to the generators of Lie algebra and optimal system

Operator Approximate Invariants

V11199051199093 1199092119906 minus 120576(1205721199093

2 )V2 119905 1199096119905 + 119906 + 120576 (minus1205721199092 + 119905)V3 119905 119906V4 119909 119906V5 119905 1199096119905 + 119906V6 119905 119906V7 119909 119906V1 + 119887V6 119905

1199093 + 120576(119887119909minus4119905) 1199092119906 minus 120576(12057211990932 + 2119887119909

3119906 )V3 + 119887V7 119905 minus 120576(119887119909) 119906V4 + 119887V2 + 119888V5 + 119889V7 1199052

2 + 1199096119887 + 120576 (( 119888119887 minus 119889) 119905) 119887119906 minus 119905 + 120576(31205721199052

2 + 119889119905)V2 + 119887V5 + 119888V7 119905 minus 120576 (119906119888 )

1199096119905 + 119906 + 120576(3120572119905119906 minus 1198881199092

721199053 )V5 + 119887V3 + 119888V7 119905 minus 120576 (119888119909119887 ) 119906

of the unperturbed equation 119906119905 minus 6119906119906119909 + 119906119909119909119909 = 0 Here 120577 =(119909 minus 119905)2 is an invariant of the group with the generator

1198830 = 120597120597119905 +

120597120597119909 (46)

The function 119908 given by (45) is invariant under the operator(46)

Using the generators 1198831 = 120597120597119909 admitted by 119906119905 minus 6119906119906119909 +119906119909119909119909 = 0 we will take the approximate symmetry

119883 = 1198830 + 1205761198831 = 120597120597119905 +

120597120597119909 + 120576 120597

120597119909 (47)

and use it looking for the approximately invariant solution of(1) in the form

119906 = 119908 + 120576V (119905 119909) (48)

Then the invariant equation test119883(119906 minus 119908 minus 120576V)|(48) = 119900(120576)can be written as

1198830 (119906 minus 119908) + 120576 [1198831 (119906 minus 119908) minus 1198830 (V)])119906=119908

= 0 (49)

Note that 1198830 does not contain the differentiation in 119906 there-fore (49) becomes

[1198831 (119906 minus 119908) minus 1198830 (V)]119906=119908

= 0 (50)

whence we obtain the following differential for V(119905 119909)V119905 + V119909 = minus (119908)119909 (51)

Because 119908 = 119908(120577) so (51) can be integrated by the variables

120577 = 119909 minus 1199052

120588 = 119905(52)

Then denoting by1199081015840 the derivative of119908with respect to 120577 wehave

(119908)119905 = minus12 (119908)1015840 (119908)119909 = 1

2 (119908)1015840(53)

and (51) becomes

V120588 = minus12 (119908)1015840 (54)

The integration yields

V = minus12120588 (119908)1015840 + 120595 (119909) (55)

Returning to the variables 119905 119909 we haveV = minus119905 (119908)119909 + 120595 (119909) (56)

Inserting this V in (48) and substituting it into (1) we obtain120595 = ((minus1 minus 3120573)6)119909Thus the approximate symmetry (47) provides the fol-

lowing approximately invariant solution (approximate trav-elling wave)

119906 (119909 119905) = minus12 sdot 1

cosh2 ((119909 minus 119905) 2)+ 120576 [minus 119905

21

cosh2 ((119909 minus 119905) 2) sdot tanh (119909 minus 1199052 )

+ minus1 minus 31205736 119909]

(57)

8 Advances in Mathematical Physics

5 Approximate Conservation Laws

Approximate Lie symmetry can be used to construct theapproximate conservation laws but in this section wewill usepartial Lagrangians to construct approximate conservationlaws of (1) this is a more concise method

Definition 9 (see [10]) An operator 119884 is a kth-order approx-imate Lie-Backlund symmetry

119884 = 1198840 + 1205761198841 + sdot sdot sdot + 120576119896119884119896 (58)

where

119884 = 120585119894 120597120597119909119894 + 120578119894 120597120597119906120572 119894 = 1 2 119899 (59)

and

119884119887 = 120585119894119887 120597120597119909119894 + 120578120572119887 120597

120597119906120572 + 120577120572119887119894 120597120597119906120572119894 + 12057712057211988711989411198942 12059712059711990612057211989411198942 + sdot sdot sdot 119887 = 0 119896

(60)

where 120585119894119887 120578120572119887 isin 119860 and the additional coefficients are

120577120572119887119894 = 119863119894 (119882120572119887 ) + 12058511989511988711990612057211989411989512057712057211988711989411198942 = 11986311989411198631198942 (119882120572119887 ) + 120585119895

11988711990612057211989511989411198942 119894 119895 = 1 119899 (61)

and119882120572119887 is the Lie characteristic function defined by

119882120572119887 = 120578120572119887 minus 120585119895119887119906120572119895 (62)

Definition 10 (see [10]) The approximate Noether operatorassociated with an approximate Lie-Backlund symmetryoperator 119884 is given by

119873119894 = 120585119894 +119882120572 120575120575119906120572 +sum

119904ge1

1198631198941 119863119894s (119882120572) 12057512057511990612057211989411989411198942 119894119904 119894 = 1 119899

(63)

where

119873119894 = 1198731198940 + 1205761198731198941 + sdot sdot sdot + 120576119896119873119894119896 119894 = 1 119899 (64)

where119863119894 is the total derivative operator defined as

119863119894 = 120597120597119909119894 + 119906119894 120597120597119906 + 119906119894119895 120597

120597119906119895 + sdot sdot sdot + 11990611989411989411198942119894119899 12059712059711990611989411198942119894119899

+ sdot sdot sdot 119894 = 1 2 119899(65)

and here119873119894119887 119887 = 0 119896 areNoether operators and the Euler-Lagrange operators are

120575120575119906120572 =

120597120597119906120572 +sum

119904ge1

(minus1)1199041198631198951 119863119895s 12059712059711990612057211989411989511198952119895119904

120572 = 1 119898 119894 = 1 119899(66)

The Euler-Lagrange approximate Lie-Backlund and approx-imate Noether operators are connected by the operatoridentity

119884 + 119863119894 (120585119894) = 119882120572 120575120575119906120572 + 119863119894119873119894 (67)

Definition 11 (see [10]) If there exists a function 119871 = 119871(119909 119906119906(1) 119906(119897) isin 119860 119897 lt 119903 and nonzero functions 119891120573120574 isin 119860 suchthat (1) which can be written as 120575119871120575119906120573 = 120576119891120573120574 1198641205741 where119891120573120574 = 119891120573120574 (119909 119906 119906(1) 119906(119903minus1) 120573 120574 = 1 119898 is an invertiblematrix then provided that 1198641205741 = 0 some 119871 is called a partialLagrangian otherwise it is the standard Lagrangian We termdifferential equations of the form

120575119871120575119906120573 = 120576119891120573120574 1198641205741 (68)

approximate Euler-Lagrange-type equations

Definition 12 (see [5]) An approximate Lie-Backlund sym-metry operator 119884 is called an approximate Noether-typesymmetry operator corresponding to a partial Lagrangian119871 isin119860 if and only if there exists a vector 119861 = (1198611 1198612 119861119899) 119861119894 isin119860 defined by

119861119894 = 1198611198940 + 1205761198611198941 + sdot sdot sdot + 120576119896119861119894119896 (69)

such that

119884 (119871) + 119871119863119894 (120585119894) = 119882120572 120575119871120575119906120572 + 119863119894 (119861119894) + 119900 (120576119896+1) (70)

where 119882 = (11988211198822 119882119898)119882120573 isin 119860 of 119884 is also thecharacteristic of the conservation law 119863119894119879119894 = 119900(120576119896+1) where

119879119894 = 119861119894 minus 119871120585119894 minus119882120573 120597119871120597119906120573119894 + sdot sdot sdot + 119900 (120576119896+1) (71)

of the approximate Euler-Lagrange-type (68)

Because (1) does not have a Lagrange function and if weput a transform 119906 = V119909 then (1) becomes

V119909119905 minus 6V119909V119909119909 + V119909119909119909119909 + 120576120572 (V119909 + 2119909V119909119909)minus 120576120573 (2V119909 + 119909V119909119909) = 0 (72)

In order to write convenient we can get

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 + 120576120572 (119906119909 + 2119909119906119909119909)minus 120576120573 (2119906119909 + 119909119906119909119909) = 0 (73)

Obviously

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 = 0 (74)

and the Lagrange function is

119871 = 12 (119906119909119909)2 + (119906119909)3 minus 1

2119906119909119906119905 (75)

Advances in Mathematical Physics 9

And the approximate 119864119906119897119890119903 minus 119871119886119892119903119886119899119892119890 minus 119905119910119901119890 equation is

120575119871120575119906 = minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909) (76)

So the approximate Noether symmetry operator is 1198840 + 1205981198841for 119871

(1198840 + 1205761198841) 119871 + 119863119894 (1205851198940 + 1205761205851198941) 119871= [(1205780 minus 1205851198950119906119895) + 120576 (1205781 minus 1205851198951119906119895)]sdot [minus120598120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 119863119894 (1198611198940 + 1205761198611198941)

(77)

where

1198840 + 1205761198841 = (12058510 + 12057612058511) 120597120597119905 + (12058520 + 12057612058521) 120597

120597119909 + (1205780+ 1205761205781) 120597

120597119906 + 120577119905 120597120597119906119905 + 120577119909 120597120597119906119909 + 120577119909119909 120597

120597119906119909119909 (78)

120577119905 = 1205780119905 + (1205780119906 minus 12058510119905) 119906119905 minus 12058520119905119906119909 minus 120585101199061199062119905 minus 12058520119906119906119909119906119905+ 12058510119906119905119905 + 12058520119906119909119905 + 120576 [1205781119905 + (1205781119906 minus 12058511119905) 119906119905 minus 12058521119905119906119909minus 120585111199061199062119905 minus 120585211199061199062119909 + 12058511119906119905119905 + 12058521119906119909119905]

(79)

120577119909 = 1205780119909 + (1205780119906 minus 12058520119909) 119906119909 minus 12058510119909119906119905 minus 120585201199061199062119909 minus 12058510119906119906119909119906119905+ 12058510119906119909119905 + 12058520119906119909119909 + 120576 [1205781119909 + (1205781119906 minus 12058521119909) 119906119909 minus 12058511119909119906119905minus 120585211199061199062119909 minus 12058521119906119906119909119906119905 + 12058511119906119909119911119905 + 12058521119906119909119909]

(80)

120577119909119909 = 1205780119909119909 + (21205780119909119906 minus 12058520119909119909) 119906119909 + (1205780119906119906 minus 212058520119909119906) 1199062119909minus 12058510119909119909119906119905 minus 120585101199061199061199062119909119906119905 minus 212058510119909119906119906119909119906119905 minus 12058510119906119906119909119909119906119905minus 212058510119909119906119909119905 minus 212058510119906119906119909119906119909119905 minus 120585201199061199061199063119909 minus 312058520119906119906119909119906119909119909+ (1205780119906 minus 212058520119909) 119906119909119909 + 12058510119906119909119909119905 + 12058520119906119909119909119909 + 120576 [1205781119909119909+ (21205781119909119906 minus 12058521119909119909) 119906119909 + (1205781119906119906 minus 212058521119909119906) 1199062119909 minus 12058511119909119909119906119905minus 120585111199061199061199062119909119906119905 minus 212058511119909119906119906119909119906119905 minus 12058511119906119906119909119909119906119905 minus 212058511119909119906119909119905minus 212058511119906119906119909119906119909119905 minus 120585211199061199061199063119909 minus 312058521119906119906119909119906119909119909+ (1205781119906 minus 212058521119909) 119906119909119909 + 12058511119906119909119909119905 + 12058521119906119909119909119909]

(81)

So (58) becomes

minus 12119906119909120577119905 minus

12119906119905120577119909 + 31199062119909120577119909 + 119906119909119909120577119909119909

+ 119871 [12058510119905 + 12058510119906119906119905 + 120576 (12058511119905 + 12058511119906119906119905)]+ 119871 [12058520119905 + 12058520119906119906119905 + 120576 (12058521119905 + 12058521119906119906119905)]= [1205780 minus 12058510119906119905 minus 12058520119906119909 + 120576 (1205781 minus 12058511119906119905 minus 12058521119906119909)]

sdot [minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 120576 (11986111119905 + 11986111119906119906119905) + 120576 (11986121119905 + 11986121119906119906119909)+ (11986110119905 + 11986110119906119906119905) + (11986120119909 + 11986120119906119906119909)

(82)

Put 119871 120577119905 120577119909 120577119909119909 into (82) and let 1205762 = 0 We obtain

minus121205780119905 minus 11986120119906 = 01205780119909 = 01205780119906 = 012058510 = 012058520 = 0

11986110119906 = 011986120119909 + 11986110119905 = 0

minus121205781119909 minus 11986111119906 = 01205781119909 = 01205781119906 = 012058511 = 012058521 = 0

11986110119906 = 011986121119909 + 11986111119905 = 0

minus121205781119905 + (120572 minus 2120573) 1205780 minus 11986121119906 = 01205781119909119909 + (2120572 minus 120573) 1205780 sdot 119909 = 0

(83)

In the following we will consider three cases of 120572 and 120573First Case 120572 = 2120573 and 120573 = 2120572

12058510 = 012058520 = 01205780 = 012058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = 119891 (119905 119909)

10 Advances in Mathematical Physics

11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905) 1199062 + 119895 (119905 119909)

(84)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909)119891(119905 119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 and 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetric operators

119884 = 120576119892 (119905) 120597120597119906 (85)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12120576119906119909119892 (119905)

1198792 = 119891 (119905 119909) + 120576 [minus12119892119905 (119905) 119906 + 119895 (119905 119909)]minus [120576119892 (119905)] [31199062119909 minus 1

2119906119905] + 119863119909 [120576119892 (119905)] 119906119909119909(86)

Second Case 120572 = 2120573 and 120573 = 212057212058510 = 012058520 = 01205780 = ℎ (119905) 12058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = minusℎ119905 (119905) 1199062 + 119891 (119905 119909) 11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905)2 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)

(87)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909) 119891(119905119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetry operators

119884 = ℎ (119905) 120597120597119906 + 120576119892 (119905) 120597

120597119906 (88)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12119906119909 [ℎ (119905) + 120576119892 (119905)]

1198792 = 119891 (119905 119909) + [minus12ℎ119905 (119905) 119906]+ 120576 [minus12119892119905 (119905) 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)]minus [120576119892 (119905) + ℎ (119905)] [31199062119909 minus 1

2119906119905]+ 119863119909 [120576119892 (119905) + ℎ (119905)] 119906119909119909

(89)

Third Case 120572 = 2120573 and 120573 = 2120572 the conservation laws are thesame as the first case

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Thisworkwas supported in part by theNatural Science Foun-dation of Inner Mongolia of China under grant 2016MS0116

References

[1] A Wazwaz Partial Differential Equations and Solitary WavesTheory Higher Education Press Beijing China 2009

[2] Y S Kivshar and B A Malomed ldquoDynamics of solitons innearly integrable systemsrdquo Reviews of Modern Physics vol 61no 4 pp 763ndash915 1989

[3] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMatematicheskii Sbornik vol 64 no 2 p 427 1989

[4] V A Bailov R K Gazizov and N H Ibragmov ldquoPertubationmethods in group analysisrdquo Journal of Mathematical Sciencesvol 55 no 1 pp 1450ndash1485 1991

[5] N H Ibragimov and V F Kovalev Approximate and Renor-mgroup Symmetries Higher Education Press Beijing China2009

[6] V A Baikov and N H Ibragimov ldquoContinuation of approxi-mate transformation groups via multiple time scales methodrdquoNonlinear Dynamics vol 22 no 1 pp 3ndash13 2000

[7] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations vol 168 ofApplied Mathematical Sciences Springer New York NY USA2010

[9] A H Kara F M Mahomed and G Unal ldquoApproximate sym-metries and conservation laws with applicationsrdquo InternationalJournal ofTheoretical Physics vol 38 no 9 pp 2389ndash2399 1999

[10] A G Johnpillai A H Kara and FMMahomed ldquoApproximateNoether-type symmetries and conservation laws via partial

Advances in Mathematical Physics 11

Lagrangians for PDEs with a small parameterrdquo Journal of Com-putational and AppliedMathematics vol 223 no 1 pp 508ndash5182009

[11] J-R Su S-L Zhang and J-N Li ldquoApproximate noether-typesymmetries and conservation laws via partial lagrangians fornonlinear wave equation with dampingrdquo Communications inTheoretical Physics vol 53 no 1 pp 37ndash42 2010

[12] E S Benilov and B A Malomed ldquoOn the spectral singularityproblem in the perturbation theory for systems close to theKdVequationrdquo Journal of Physics A Mathematical and General vol21 no 8 pp L439ndashL442 1988

[13] M A Allen and G Rowlands ldquoA solitary-wave solution to aperturbedKdV equationrdquo Journal of Plasma Physics vol 64 no4 pp 475ndash480 2000

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Advances in Mathematical Physics 7

Table 5 Approximate invariants with respect to the generators of Lie algebra and optimal system

Operator Approximate Invariants

V11199051199093 1199092119906 minus 120576(1205721199093

2 )V2 119905 1199096119905 + 119906 + 120576 (minus1205721199092 + 119905)V3 119905 119906V4 119909 119906V5 119905 1199096119905 + 119906V6 119905 119906V7 119909 119906V1 + 119887V6 119905

1199093 + 120576(119887119909minus4119905) 1199092119906 minus 120576(12057211990932 + 2119887119909

3119906 )V3 + 119887V7 119905 minus 120576(119887119909) 119906V4 + 119887V2 + 119888V5 + 119889V7 1199052

2 + 1199096119887 + 120576 (( 119888119887 minus 119889) 119905) 119887119906 minus 119905 + 120576(31205721199052

2 + 119889119905)V2 + 119887V5 + 119888V7 119905 minus 120576 (119906119888 )

1199096119905 + 119906 + 120576(3120572119905119906 minus 1198881199092

721199053 )V5 + 119887V3 + 119888V7 119905 minus 120576 (119888119909119887 ) 119906

of the unperturbed equation 119906119905 minus 6119906119906119909 + 119906119909119909119909 = 0 Here 120577 =(119909 minus 119905)2 is an invariant of the group with the generator

1198830 = 120597120597119905 +

120597120597119909 (46)

The function 119908 given by (45) is invariant under the operator(46)

Using the generators 1198831 = 120597120597119909 admitted by 119906119905 minus 6119906119906119909 +119906119909119909119909 = 0 we will take the approximate symmetry

119883 = 1198830 + 1205761198831 = 120597120597119905 +

120597120597119909 + 120576 120597

120597119909 (47)

and use it looking for the approximately invariant solution of(1) in the form

119906 = 119908 + 120576V (119905 119909) (48)

Then the invariant equation test119883(119906 minus 119908 minus 120576V)|(48) = 119900(120576)can be written as

1198830 (119906 minus 119908) + 120576 [1198831 (119906 minus 119908) minus 1198830 (V)])119906=119908

= 0 (49)

Note that 1198830 does not contain the differentiation in 119906 there-fore (49) becomes

[1198831 (119906 minus 119908) minus 1198830 (V)]119906=119908

= 0 (50)

whence we obtain the following differential for V(119905 119909)V119905 + V119909 = minus (119908)119909 (51)

Because 119908 = 119908(120577) so (51) can be integrated by the variables

120577 = 119909 minus 1199052

120588 = 119905(52)

Then denoting by1199081015840 the derivative of119908with respect to 120577 wehave

(119908)119905 = minus12 (119908)1015840 (119908)119909 = 1

2 (119908)1015840(53)

and (51) becomes

V120588 = minus12 (119908)1015840 (54)

The integration yields

V = minus12120588 (119908)1015840 + 120595 (119909) (55)

Returning to the variables 119905 119909 we haveV = minus119905 (119908)119909 + 120595 (119909) (56)

Inserting this V in (48) and substituting it into (1) we obtain120595 = ((minus1 minus 3120573)6)119909Thus the approximate symmetry (47) provides the fol-

lowing approximately invariant solution (approximate trav-elling wave)

119906 (119909 119905) = minus12 sdot 1

cosh2 ((119909 minus 119905) 2)+ 120576 [minus 119905

21

cosh2 ((119909 minus 119905) 2) sdot tanh (119909 minus 1199052 )

+ minus1 minus 31205736 119909]

(57)

8 Advances in Mathematical Physics

5 Approximate Conservation Laws

Approximate Lie symmetry can be used to construct theapproximate conservation laws but in this section wewill usepartial Lagrangians to construct approximate conservationlaws of (1) this is a more concise method

Definition 9 (see [10]) An operator 119884 is a kth-order approx-imate Lie-Backlund symmetry

119884 = 1198840 + 1205761198841 + sdot sdot sdot + 120576119896119884119896 (58)

where

119884 = 120585119894 120597120597119909119894 + 120578119894 120597120597119906120572 119894 = 1 2 119899 (59)

and

119884119887 = 120585119894119887 120597120597119909119894 + 120578120572119887 120597

120597119906120572 + 120577120572119887119894 120597120597119906120572119894 + 12057712057211988711989411198942 12059712059711990612057211989411198942 + sdot sdot sdot 119887 = 0 119896

(60)

where 120585119894119887 120578120572119887 isin 119860 and the additional coefficients are

120577120572119887119894 = 119863119894 (119882120572119887 ) + 12058511989511988711990612057211989411989512057712057211988711989411198942 = 11986311989411198631198942 (119882120572119887 ) + 120585119895

11988711990612057211989511989411198942 119894 119895 = 1 119899 (61)

and119882120572119887 is the Lie characteristic function defined by

119882120572119887 = 120578120572119887 minus 120585119895119887119906120572119895 (62)

Definition 10 (see [10]) The approximate Noether operatorassociated with an approximate Lie-Backlund symmetryoperator 119884 is given by

119873119894 = 120585119894 +119882120572 120575120575119906120572 +sum

119904ge1

1198631198941 119863119894s (119882120572) 12057512057511990612057211989411989411198942 119894119904 119894 = 1 119899

(63)

where

119873119894 = 1198731198940 + 1205761198731198941 + sdot sdot sdot + 120576119896119873119894119896 119894 = 1 119899 (64)

where119863119894 is the total derivative operator defined as

119863119894 = 120597120597119909119894 + 119906119894 120597120597119906 + 119906119894119895 120597

120597119906119895 + sdot sdot sdot + 11990611989411989411198942119894119899 12059712059711990611989411198942119894119899

+ sdot sdot sdot 119894 = 1 2 119899(65)

and here119873119894119887 119887 = 0 119896 areNoether operators and the Euler-Lagrange operators are

120575120575119906120572 =

120597120597119906120572 +sum

119904ge1

(minus1)1199041198631198951 119863119895s 12059712059711990612057211989411989511198952119895119904

120572 = 1 119898 119894 = 1 119899(66)

The Euler-Lagrange approximate Lie-Backlund and approx-imate Noether operators are connected by the operatoridentity

119884 + 119863119894 (120585119894) = 119882120572 120575120575119906120572 + 119863119894119873119894 (67)

Definition 11 (see [10]) If there exists a function 119871 = 119871(119909 119906119906(1) 119906(119897) isin 119860 119897 lt 119903 and nonzero functions 119891120573120574 isin 119860 suchthat (1) which can be written as 120575119871120575119906120573 = 120576119891120573120574 1198641205741 where119891120573120574 = 119891120573120574 (119909 119906 119906(1) 119906(119903minus1) 120573 120574 = 1 119898 is an invertiblematrix then provided that 1198641205741 = 0 some 119871 is called a partialLagrangian otherwise it is the standard Lagrangian We termdifferential equations of the form

120575119871120575119906120573 = 120576119891120573120574 1198641205741 (68)

approximate Euler-Lagrange-type equations

Definition 12 (see [5]) An approximate Lie-Backlund sym-metry operator 119884 is called an approximate Noether-typesymmetry operator corresponding to a partial Lagrangian119871 isin119860 if and only if there exists a vector 119861 = (1198611 1198612 119861119899) 119861119894 isin119860 defined by

119861119894 = 1198611198940 + 1205761198611198941 + sdot sdot sdot + 120576119896119861119894119896 (69)

such that

119884 (119871) + 119871119863119894 (120585119894) = 119882120572 120575119871120575119906120572 + 119863119894 (119861119894) + 119900 (120576119896+1) (70)

where 119882 = (11988211198822 119882119898)119882120573 isin 119860 of 119884 is also thecharacteristic of the conservation law 119863119894119879119894 = 119900(120576119896+1) where

119879119894 = 119861119894 minus 119871120585119894 minus119882120573 120597119871120597119906120573119894 + sdot sdot sdot + 119900 (120576119896+1) (71)

of the approximate Euler-Lagrange-type (68)

Because (1) does not have a Lagrange function and if weput a transform 119906 = V119909 then (1) becomes

V119909119905 minus 6V119909V119909119909 + V119909119909119909119909 + 120576120572 (V119909 + 2119909V119909119909)minus 120576120573 (2V119909 + 119909V119909119909) = 0 (72)

In order to write convenient we can get

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 + 120576120572 (119906119909 + 2119909119906119909119909)minus 120576120573 (2119906119909 + 119909119906119909119909) = 0 (73)

Obviously

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 = 0 (74)

and the Lagrange function is

119871 = 12 (119906119909119909)2 + (119906119909)3 minus 1

2119906119909119906119905 (75)

Advances in Mathematical Physics 9

And the approximate 119864119906119897119890119903 minus 119871119886119892119903119886119899119892119890 minus 119905119910119901119890 equation is

120575119871120575119906 = minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909) (76)

So the approximate Noether symmetry operator is 1198840 + 1205981198841for 119871

(1198840 + 1205761198841) 119871 + 119863119894 (1205851198940 + 1205761205851198941) 119871= [(1205780 minus 1205851198950119906119895) + 120576 (1205781 minus 1205851198951119906119895)]sdot [minus120598120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 119863119894 (1198611198940 + 1205761198611198941)

(77)

where

1198840 + 1205761198841 = (12058510 + 12057612058511) 120597120597119905 + (12058520 + 12057612058521) 120597

120597119909 + (1205780+ 1205761205781) 120597

120597119906 + 120577119905 120597120597119906119905 + 120577119909 120597120597119906119909 + 120577119909119909 120597

120597119906119909119909 (78)

120577119905 = 1205780119905 + (1205780119906 minus 12058510119905) 119906119905 minus 12058520119905119906119909 minus 120585101199061199062119905 minus 12058520119906119906119909119906119905+ 12058510119906119905119905 + 12058520119906119909119905 + 120576 [1205781119905 + (1205781119906 minus 12058511119905) 119906119905 minus 12058521119905119906119909minus 120585111199061199062119905 minus 120585211199061199062119909 + 12058511119906119905119905 + 12058521119906119909119905]

(79)

120577119909 = 1205780119909 + (1205780119906 minus 12058520119909) 119906119909 minus 12058510119909119906119905 minus 120585201199061199062119909 minus 12058510119906119906119909119906119905+ 12058510119906119909119905 + 12058520119906119909119909 + 120576 [1205781119909 + (1205781119906 minus 12058521119909) 119906119909 minus 12058511119909119906119905minus 120585211199061199062119909 minus 12058521119906119906119909119906119905 + 12058511119906119909119911119905 + 12058521119906119909119909]

(80)

120577119909119909 = 1205780119909119909 + (21205780119909119906 minus 12058520119909119909) 119906119909 + (1205780119906119906 minus 212058520119909119906) 1199062119909minus 12058510119909119909119906119905 minus 120585101199061199061199062119909119906119905 minus 212058510119909119906119906119909119906119905 minus 12058510119906119906119909119909119906119905minus 212058510119909119906119909119905 minus 212058510119906119906119909119906119909119905 minus 120585201199061199061199063119909 minus 312058520119906119906119909119906119909119909+ (1205780119906 minus 212058520119909) 119906119909119909 + 12058510119906119909119909119905 + 12058520119906119909119909119909 + 120576 [1205781119909119909+ (21205781119909119906 minus 12058521119909119909) 119906119909 + (1205781119906119906 minus 212058521119909119906) 1199062119909 minus 12058511119909119909119906119905minus 120585111199061199061199062119909119906119905 minus 212058511119909119906119906119909119906119905 minus 12058511119906119906119909119909119906119905 minus 212058511119909119906119909119905minus 212058511119906119906119909119906119909119905 minus 120585211199061199061199063119909 minus 312058521119906119906119909119906119909119909+ (1205781119906 minus 212058521119909) 119906119909119909 + 12058511119906119909119909119905 + 12058521119906119909119909119909]

(81)

So (58) becomes

minus 12119906119909120577119905 minus

12119906119905120577119909 + 31199062119909120577119909 + 119906119909119909120577119909119909

+ 119871 [12058510119905 + 12058510119906119906119905 + 120576 (12058511119905 + 12058511119906119906119905)]+ 119871 [12058520119905 + 12058520119906119906119905 + 120576 (12058521119905 + 12058521119906119906119905)]= [1205780 minus 12058510119906119905 minus 12058520119906119909 + 120576 (1205781 minus 12058511119906119905 minus 12058521119906119909)]

sdot [minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 120576 (11986111119905 + 11986111119906119906119905) + 120576 (11986121119905 + 11986121119906119906119909)+ (11986110119905 + 11986110119906119906119905) + (11986120119909 + 11986120119906119906119909)

(82)

Put 119871 120577119905 120577119909 120577119909119909 into (82) and let 1205762 = 0 We obtain

minus121205780119905 minus 11986120119906 = 01205780119909 = 01205780119906 = 012058510 = 012058520 = 0

11986110119906 = 011986120119909 + 11986110119905 = 0

minus121205781119909 minus 11986111119906 = 01205781119909 = 01205781119906 = 012058511 = 012058521 = 0

11986110119906 = 011986121119909 + 11986111119905 = 0

minus121205781119905 + (120572 minus 2120573) 1205780 minus 11986121119906 = 01205781119909119909 + (2120572 minus 120573) 1205780 sdot 119909 = 0

(83)

In the following we will consider three cases of 120572 and 120573First Case 120572 = 2120573 and 120573 = 2120572

12058510 = 012058520 = 01205780 = 012058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = 119891 (119905 119909)

10 Advances in Mathematical Physics

11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905) 1199062 + 119895 (119905 119909)

(84)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909)119891(119905 119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 and 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetric operators

119884 = 120576119892 (119905) 120597120597119906 (85)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12120576119906119909119892 (119905)

1198792 = 119891 (119905 119909) + 120576 [minus12119892119905 (119905) 119906 + 119895 (119905 119909)]minus [120576119892 (119905)] [31199062119909 minus 1

2119906119905] + 119863119909 [120576119892 (119905)] 119906119909119909(86)

Second Case 120572 = 2120573 and 120573 = 212057212058510 = 012058520 = 01205780 = ℎ (119905) 12058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = minusℎ119905 (119905) 1199062 + 119891 (119905 119909) 11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905)2 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)

(87)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909) 119891(119905119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetry operators

119884 = ℎ (119905) 120597120597119906 + 120576119892 (119905) 120597

120597119906 (88)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12119906119909 [ℎ (119905) + 120576119892 (119905)]

1198792 = 119891 (119905 119909) + [minus12ℎ119905 (119905) 119906]+ 120576 [minus12119892119905 (119905) 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)]minus [120576119892 (119905) + ℎ (119905)] [31199062119909 minus 1

2119906119905]+ 119863119909 [120576119892 (119905) + ℎ (119905)] 119906119909119909

(89)

Third Case 120572 = 2120573 and 120573 = 2120572 the conservation laws are thesame as the first case

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Thisworkwas supported in part by theNatural Science Foun-dation of Inner Mongolia of China under grant 2016MS0116

References

[1] A Wazwaz Partial Differential Equations and Solitary WavesTheory Higher Education Press Beijing China 2009

[2] Y S Kivshar and B A Malomed ldquoDynamics of solitons innearly integrable systemsrdquo Reviews of Modern Physics vol 61no 4 pp 763ndash915 1989

[3] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMatematicheskii Sbornik vol 64 no 2 p 427 1989

[4] V A Bailov R K Gazizov and N H Ibragmov ldquoPertubationmethods in group analysisrdquo Journal of Mathematical Sciencesvol 55 no 1 pp 1450ndash1485 1991

[5] N H Ibragimov and V F Kovalev Approximate and Renor-mgroup Symmetries Higher Education Press Beijing China2009

[6] V A Baikov and N H Ibragimov ldquoContinuation of approxi-mate transformation groups via multiple time scales methodrdquoNonlinear Dynamics vol 22 no 1 pp 3ndash13 2000

[7] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations vol 168 ofApplied Mathematical Sciences Springer New York NY USA2010

[9] A H Kara F M Mahomed and G Unal ldquoApproximate sym-metries and conservation laws with applicationsrdquo InternationalJournal ofTheoretical Physics vol 38 no 9 pp 2389ndash2399 1999

[10] A G Johnpillai A H Kara and FMMahomed ldquoApproximateNoether-type symmetries and conservation laws via partial

Advances in Mathematical Physics 11

Lagrangians for PDEs with a small parameterrdquo Journal of Com-putational and AppliedMathematics vol 223 no 1 pp 508ndash5182009

[11] J-R Su S-L Zhang and J-N Li ldquoApproximate noether-typesymmetries and conservation laws via partial lagrangians fornonlinear wave equation with dampingrdquo Communications inTheoretical Physics vol 53 no 1 pp 37ndash42 2010

[12] E S Benilov and B A Malomed ldquoOn the spectral singularityproblem in the perturbation theory for systems close to theKdVequationrdquo Journal of Physics A Mathematical and General vol21 no 8 pp L439ndashL442 1988

[13] M A Allen and G Rowlands ldquoA solitary-wave solution to aperturbedKdV equationrdquo Journal of Plasma Physics vol 64 no4 pp 475ndash480 2000

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Submit your manuscripts atwwwhindawicom

8 Advances in Mathematical Physics

5 Approximate Conservation Laws

Approximate Lie symmetry can be used to construct theapproximate conservation laws but in this section wewill usepartial Lagrangians to construct approximate conservationlaws of (1) this is a more concise method

Definition 9 (see [10]) An operator 119884 is a kth-order approx-imate Lie-Backlund symmetry

119884 = 1198840 + 1205761198841 + sdot sdot sdot + 120576119896119884119896 (58)

where

119884 = 120585119894 120597120597119909119894 + 120578119894 120597120597119906120572 119894 = 1 2 119899 (59)

and

119884119887 = 120585119894119887 120597120597119909119894 + 120578120572119887 120597

120597119906120572 + 120577120572119887119894 120597120597119906120572119894 + 12057712057211988711989411198942 12059712059711990612057211989411198942 + sdot sdot sdot 119887 = 0 119896

(60)

where 120585119894119887 120578120572119887 isin 119860 and the additional coefficients are

120577120572119887119894 = 119863119894 (119882120572119887 ) + 12058511989511988711990612057211989411989512057712057211988711989411198942 = 11986311989411198631198942 (119882120572119887 ) + 120585119895

11988711990612057211989511989411198942 119894 119895 = 1 119899 (61)

and119882120572119887 is the Lie characteristic function defined by

119882120572119887 = 120578120572119887 minus 120585119895119887119906120572119895 (62)

Definition 10 (see [10]) The approximate Noether operatorassociated with an approximate Lie-Backlund symmetryoperator 119884 is given by

119873119894 = 120585119894 +119882120572 120575120575119906120572 +sum

119904ge1

1198631198941 119863119894s (119882120572) 12057512057511990612057211989411989411198942 119894119904 119894 = 1 119899

(63)

where

119873119894 = 1198731198940 + 1205761198731198941 + sdot sdot sdot + 120576119896119873119894119896 119894 = 1 119899 (64)

where119863119894 is the total derivative operator defined as

119863119894 = 120597120597119909119894 + 119906119894 120597120597119906 + 119906119894119895 120597

120597119906119895 + sdot sdot sdot + 11990611989411989411198942119894119899 12059712059711990611989411198942119894119899

+ sdot sdot sdot 119894 = 1 2 119899(65)

and here119873119894119887 119887 = 0 119896 areNoether operators and the Euler-Lagrange operators are

120575120575119906120572 =

120597120597119906120572 +sum

119904ge1

(minus1)1199041198631198951 119863119895s 12059712059711990612057211989411989511198952119895119904

120572 = 1 119898 119894 = 1 119899(66)

The Euler-Lagrange approximate Lie-Backlund and approx-imate Noether operators are connected by the operatoridentity

119884 + 119863119894 (120585119894) = 119882120572 120575120575119906120572 + 119863119894119873119894 (67)

Definition 11 (see [10]) If there exists a function 119871 = 119871(119909 119906119906(1) 119906(119897) isin 119860 119897 lt 119903 and nonzero functions 119891120573120574 isin 119860 suchthat (1) which can be written as 120575119871120575119906120573 = 120576119891120573120574 1198641205741 where119891120573120574 = 119891120573120574 (119909 119906 119906(1) 119906(119903minus1) 120573 120574 = 1 119898 is an invertiblematrix then provided that 1198641205741 = 0 some 119871 is called a partialLagrangian otherwise it is the standard Lagrangian We termdifferential equations of the form

120575119871120575119906120573 = 120576119891120573120574 1198641205741 (68)

approximate Euler-Lagrange-type equations

Definition 12 (see [5]) An approximate Lie-Backlund sym-metry operator 119884 is called an approximate Noether-typesymmetry operator corresponding to a partial Lagrangian119871 isin119860 if and only if there exists a vector 119861 = (1198611 1198612 119861119899) 119861119894 isin119860 defined by

119861119894 = 1198611198940 + 1205761198611198941 + sdot sdot sdot + 120576119896119861119894119896 (69)

such that

119884 (119871) + 119871119863119894 (120585119894) = 119882120572 120575119871120575119906120572 + 119863119894 (119861119894) + 119900 (120576119896+1) (70)

where 119882 = (11988211198822 119882119898)119882120573 isin 119860 of 119884 is also thecharacteristic of the conservation law 119863119894119879119894 = 119900(120576119896+1) where

119879119894 = 119861119894 minus 119871120585119894 minus119882120573 120597119871120597119906120573119894 + sdot sdot sdot + 119900 (120576119896+1) (71)

of the approximate Euler-Lagrange-type (68)

Because (1) does not have a Lagrange function and if weput a transform 119906 = V119909 then (1) becomes

V119909119905 minus 6V119909V119909119909 + V119909119909119909119909 + 120576120572 (V119909 + 2119909V119909119909)minus 120576120573 (2V119909 + 119909V119909119909) = 0 (72)

In order to write convenient we can get

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 + 120576120572 (119906119909 + 2119909119906119909119909)minus 120576120573 (2119906119909 + 119909119906119909119909) = 0 (73)

Obviously

119906119909119905 minus 6119906119909119906119909119909 + 119906119909119909119909119909 = 0 (74)

and the Lagrange function is

119871 = 12 (119906119909119909)2 + (119906119909)3 minus 1

2119906119909119906119905 (75)

Advances in Mathematical Physics 9

And the approximate 119864119906119897119890119903 minus 119871119886119892119903119886119899119892119890 minus 119905119910119901119890 equation is

120575119871120575119906 = minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909) (76)

So the approximate Noether symmetry operator is 1198840 + 1205981198841for 119871

(1198840 + 1205761198841) 119871 + 119863119894 (1205851198940 + 1205761205851198941) 119871= [(1205780 minus 1205851198950119906119895) + 120576 (1205781 minus 1205851198951119906119895)]sdot [minus120598120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 119863119894 (1198611198940 + 1205761198611198941)

(77)

where

1198840 + 1205761198841 = (12058510 + 12057612058511) 120597120597119905 + (12058520 + 12057612058521) 120597

120597119909 + (1205780+ 1205761205781) 120597

120597119906 + 120577119905 120597120597119906119905 + 120577119909 120597120597119906119909 + 120577119909119909 120597

120597119906119909119909 (78)

120577119905 = 1205780119905 + (1205780119906 minus 12058510119905) 119906119905 minus 12058520119905119906119909 minus 120585101199061199062119905 minus 12058520119906119906119909119906119905+ 12058510119906119905119905 + 12058520119906119909119905 + 120576 [1205781119905 + (1205781119906 minus 12058511119905) 119906119905 minus 12058521119905119906119909minus 120585111199061199062119905 minus 120585211199061199062119909 + 12058511119906119905119905 + 12058521119906119909119905]

(79)

120577119909 = 1205780119909 + (1205780119906 minus 12058520119909) 119906119909 minus 12058510119909119906119905 minus 120585201199061199062119909 minus 12058510119906119906119909119906119905+ 12058510119906119909119905 + 12058520119906119909119909 + 120576 [1205781119909 + (1205781119906 minus 12058521119909) 119906119909 minus 12058511119909119906119905minus 120585211199061199062119909 minus 12058521119906119906119909119906119905 + 12058511119906119909119911119905 + 12058521119906119909119909]

(80)

120577119909119909 = 1205780119909119909 + (21205780119909119906 minus 12058520119909119909) 119906119909 + (1205780119906119906 minus 212058520119909119906) 1199062119909minus 12058510119909119909119906119905 minus 120585101199061199061199062119909119906119905 minus 212058510119909119906119906119909119906119905 minus 12058510119906119906119909119909119906119905minus 212058510119909119906119909119905 minus 212058510119906119906119909119906119909119905 minus 120585201199061199061199063119909 minus 312058520119906119906119909119906119909119909+ (1205780119906 minus 212058520119909) 119906119909119909 + 12058510119906119909119909119905 + 12058520119906119909119909119909 + 120576 [1205781119909119909+ (21205781119909119906 minus 12058521119909119909) 119906119909 + (1205781119906119906 minus 212058521119909119906) 1199062119909 minus 12058511119909119909119906119905minus 120585111199061199061199062119909119906119905 minus 212058511119909119906119906119909119906119905 minus 12058511119906119906119909119909119906119905 minus 212058511119909119906119909119905minus 212058511119906119906119909119906119909119905 minus 120585211199061199061199063119909 minus 312058521119906119906119909119906119909119909+ (1205781119906 minus 212058521119909) 119906119909119909 + 12058511119906119909119909119905 + 12058521119906119909119909119909]

(81)

So (58) becomes

minus 12119906119909120577119905 minus

12119906119905120577119909 + 31199062119909120577119909 + 119906119909119909120577119909119909

+ 119871 [12058510119905 + 12058510119906119906119905 + 120576 (12058511119905 + 12058511119906119906119905)]+ 119871 [12058520119905 + 12058520119906119906119905 + 120576 (12058521119905 + 12058521119906119906119905)]= [1205780 minus 12058510119906119905 minus 12058520119906119909 + 120576 (1205781 minus 12058511119906119905 minus 12058521119906119909)]

sdot [minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 120576 (11986111119905 + 11986111119906119906119905) + 120576 (11986121119905 + 11986121119906119906119909)+ (11986110119905 + 11986110119906119906119905) + (11986120119909 + 11986120119906119906119909)

(82)

Put 119871 120577119905 120577119909 120577119909119909 into (82) and let 1205762 = 0 We obtain

minus121205780119905 minus 11986120119906 = 01205780119909 = 01205780119906 = 012058510 = 012058520 = 0

11986110119906 = 011986120119909 + 11986110119905 = 0

minus121205781119909 minus 11986111119906 = 01205781119909 = 01205781119906 = 012058511 = 012058521 = 0

11986110119906 = 011986121119909 + 11986111119905 = 0

minus121205781119905 + (120572 minus 2120573) 1205780 minus 11986121119906 = 01205781119909119909 + (2120572 minus 120573) 1205780 sdot 119909 = 0

(83)

In the following we will consider three cases of 120572 and 120573First Case 120572 = 2120573 and 120573 = 2120572

12058510 = 012058520 = 01205780 = 012058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = 119891 (119905 119909)

10 Advances in Mathematical Physics

11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905) 1199062 + 119895 (119905 119909)

(84)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909)119891(119905 119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 and 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetric operators

119884 = 120576119892 (119905) 120597120597119906 (85)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12120576119906119909119892 (119905)

1198792 = 119891 (119905 119909) + 120576 [minus12119892119905 (119905) 119906 + 119895 (119905 119909)]minus [120576119892 (119905)] [31199062119909 minus 1

2119906119905] + 119863119909 [120576119892 (119905)] 119906119909119909(86)

Second Case 120572 = 2120573 and 120573 = 212057212058510 = 012058520 = 01205780 = ℎ (119905) 12058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = minusℎ119905 (119905) 1199062 + 119891 (119905 119909) 11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905)2 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)

(87)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909) 119891(119905119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetry operators

119884 = ℎ (119905) 120597120597119906 + 120576119892 (119905) 120597

120597119906 (88)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12119906119909 [ℎ (119905) + 120576119892 (119905)]

1198792 = 119891 (119905 119909) + [minus12ℎ119905 (119905) 119906]+ 120576 [minus12119892119905 (119905) 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)]minus [120576119892 (119905) + ℎ (119905)] [31199062119909 minus 1

2119906119905]+ 119863119909 [120576119892 (119905) + ℎ (119905)] 119906119909119909

(89)

Third Case 120572 = 2120573 and 120573 = 2120572 the conservation laws are thesame as the first case

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Thisworkwas supported in part by theNatural Science Foun-dation of Inner Mongolia of China under grant 2016MS0116

References

[1] A Wazwaz Partial Differential Equations and Solitary WavesTheory Higher Education Press Beijing China 2009

[2] Y S Kivshar and B A Malomed ldquoDynamics of solitons innearly integrable systemsrdquo Reviews of Modern Physics vol 61no 4 pp 763ndash915 1989

[3] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMatematicheskii Sbornik vol 64 no 2 p 427 1989

[4] V A Bailov R K Gazizov and N H Ibragmov ldquoPertubationmethods in group analysisrdquo Journal of Mathematical Sciencesvol 55 no 1 pp 1450ndash1485 1991

[5] N H Ibragimov and V F Kovalev Approximate and Renor-mgroup Symmetries Higher Education Press Beijing China2009

[6] V A Baikov and N H Ibragimov ldquoContinuation of approxi-mate transformation groups via multiple time scales methodrdquoNonlinear Dynamics vol 22 no 1 pp 3ndash13 2000

[7] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations vol 168 ofApplied Mathematical Sciences Springer New York NY USA2010

[9] A H Kara F M Mahomed and G Unal ldquoApproximate sym-metries and conservation laws with applicationsrdquo InternationalJournal ofTheoretical Physics vol 38 no 9 pp 2389ndash2399 1999

[10] A G Johnpillai A H Kara and FMMahomed ldquoApproximateNoether-type symmetries and conservation laws via partial

Advances in Mathematical Physics 11

Lagrangians for PDEs with a small parameterrdquo Journal of Com-putational and AppliedMathematics vol 223 no 1 pp 508ndash5182009

[11] J-R Su S-L Zhang and J-N Li ldquoApproximate noether-typesymmetries and conservation laws via partial lagrangians fornonlinear wave equation with dampingrdquo Communications inTheoretical Physics vol 53 no 1 pp 37ndash42 2010

[12] E S Benilov and B A Malomed ldquoOn the spectral singularityproblem in the perturbation theory for systems close to theKdVequationrdquo Journal of Physics A Mathematical and General vol21 no 8 pp L439ndashL442 1988

[13] M A Allen and G Rowlands ldquoA solitary-wave solution to aperturbedKdV equationrdquo Journal of Plasma Physics vol 64 no4 pp 475ndash480 2000

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Advances in Mathematical Physics 9

And the approximate 119864119906119897119890119903 minus 119871119886119892119903119886119899119892119890 minus 119905119910119901119890 equation is

120575119871120575119906 = minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909) (76)

So the approximate Noether symmetry operator is 1198840 + 1205981198841for 119871

(1198840 + 1205761198841) 119871 + 119863119894 (1205851198940 + 1205761205851198941) 119871= [(1205780 minus 1205851198950119906119895) + 120576 (1205781 minus 1205851198951119906119895)]sdot [minus120598120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 119863119894 (1198611198940 + 1205761198611198941)

(77)

where

1198840 + 1205761198841 = (12058510 + 12057612058511) 120597120597119905 + (12058520 + 12057612058521) 120597

120597119909 + (1205780+ 1205761205781) 120597

120597119906 + 120577119905 120597120597119906119905 + 120577119909 120597120597119906119909 + 120577119909119909 120597

120597119906119909119909 (78)

120577119905 = 1205780119905 + (1205780119906 minus 12058510119905) 119906119905 minus 12058520119905119906119909 minus 120585101199061199062119905 minus 12058520119906119906119909119906119905+ 12058510119906119905119905 + 12058520119906119909119905 + 120576 [1205781119905 + (1205781119906 minus 12058511119905) 119906119905 minus 12058521119905119906119909minus 120585111199061199062119905 minus 120585211199061199062119909 + 12058511119906119905119905 + 12058521119906119909119905]

(79)

120577119909 = 1205780119909 + (1205780119906 minus 12058520119909) 119906119909 minus 12058510119909119906119905 minus 120585201199061199062119909 minus 12058510119906119906119909119906119905+ 12058510119906119909119905 + 12058520119906119909119909 + 120576 [1205781119909 + (1205781119906 minus 12058521119909) 119906119909 minus 12058511119909119906119905minus 120585211199061199062119909 minus 12058521119906119906119909119906119905 + 12058511119906119909119911119905 + 12058521119906119909119909]

(80)

120577119909119909 = 1205780119909119909 + (21205780119909119906 minus 12058520119909119909) 119906119909 + (1205780119906119906 minus 212058520119909119906) 1199062119909minus 12058510119909119909119906119905 minus 120585101199061199061199062119909119906119905 minus 212058510119909119906119906119909119906119905 minus 12058510119906119906119909119909119906119905minus 212058510119909119906119909119905 minus 212058510119906119906119909119906119909119905 minus 120585201199061199061199063119909 minus 312058520119906119906119909119906119909119909+ (1205780119906 minus 212058520119909) 119906119909119909 + 12058510119906119909119909119905 + 12058520119906119909119909119909 + 120576 [1205781119909119909+ (21205781119909119906 minus 12058521119909119909) 119906119909 + (1205781119906119906 minus 212058521119909119906) 1199062119909 minus 12058511119909119909119906119905minus 120585111199061199061199062119909119906119905 minus 212058511119909119906119906119909119906119905 minus 12058511119906119906119909119909119906119905 minus 212058511119909119906119909119905minus 212058511119906119906119909119906119909119905 minus 120585211199061199061199063119909 minus 312058521119906119906119909119906119909119909+ (1205781119906 minus 212058521119909) 119906119909119909 + 12058511119906119909119909119905 + 12058521119906119909119909119909]

(81)

So (58) becomes

minus 12119906119909120577119905 minus

12119906119905120577119909 + 31199062119909120577119909 + 119906119909119909120577119909119909

+ 119871 [12058510119905 + 12058510119906119906119905 + 120576 (12058511119905 + 12058511119906119906119905)]+ 119871 [12058520119905 + 12058520119906119906119905 + 120576 (12058521119905 + 12058521119906119906119905)]= [1205780 minus 12058510119906119905 minus 12058520119906119909 + 120576 (1205781 minus 12058511119906119905 minus 12058521119906119909)]

sdot [minus120576120572 (119906119909 + 2119909119906119909119909) + 120576120573 (2119906119909 + 119909119906119909119909)]+ 120576 (11986111119905 + 11986111119906119906119905) + 120576 (11986121119905 + 11986121119906119906119909)+ (11986110119905 + 11986110119906119906119905) + (11986120119909 + 11986120119906119906119909)

(82)

Put 119871 120577119905 120577119909 120577119909119909 into (82) and let 1205762 = 0 We obtain

minus121205780119905 minus 11986120119906 = 01205780119909 = 01205780119906 = 012058510 = 012058520 = 0

11986110119906 = 011986120119909 + 11986110119905 = 0

minus121205781119909 minus 11986111119906 = 01205781119909 = 01205781119906 = 012058511 = 012058521 = 0

11986110119906 = 011986121119909 + 11986111119905 = 0

minus121205781119905 + (120572 minus 2120573) 1205780 minus 11986121119906 = 01205781119909119909 + (2120572 minus 120573) 1205780 sdot 119909 = 0

(83)

In the following we will consider three cases of 120572 and 120573First Case 120572 = 2120573 and 120573 = 2120572

12058510 = 012058520 = 01205780 = 012058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = 119891 (119905 119909)

10 Advances in Mathematical Physics

11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905) 1199062 + 119895 (119905 119909)

(84)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909)119891(119905 119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 and 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetric operators

119884 = 120576119892 (119905) 120597120597119906 (85)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12120576119906119909119892 (119905)

1198792 = 119891 (119905 119909) + 120576 [minus12119892119905 (119905) 119906 + 119895 (119905 119909)]minus [120576119892 (119905)] [31199062119909 minus 1

2119906119905] + 119863119909 [120576119892 (119905)] 119906119909119909(86)

Second Case 120572 = 2120573 and 120573 = 212057212058510 = 012058520 = 01205780 = ℎ (119905) 12058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = minusℎ119905 (119905) 1199062 + 119891 (119905 119909) 11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905)2 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)

(87)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909) 119891(119905119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetry operators

119884 = ℎ (119905) 120597120597119906 + 120576119892 (119905) 120597

120597119906 (88)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12119906119909 [ℎ (119905) + 120576119892 (119905)]

1198792 = 119891 (119905 119909) + [minus12ℎ119905 (119905) 119906]+ 120576 [minus12119892119905 (119905) 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)]minus [120576119892 (119905) + ℎ (119905)] [31199062119909 minus 1

2119906119905]+ 119863119909 [120576119892 (119905) + ℎ (119905)] 119906119909119909

(89)

Third Case 120572 = 2120573 and 120573 = 2120572 the conservation laws are thesame as the first case

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Thisworkwas supported in part by theNatural Science Foun-dation of Inner Mongolia of China under grant 2016MS0116

References

[1] A Wazwaz Partial Differential Equations and Solitary WavesTheory Higher Education Press Beijing China 2009

[2] Y S Kivshar and B A Malomed ldquoDynamics of solitons innearly integrable systemsrdquo Reviews of Modern Physics vol 61no 4 pp 763ndash915 1989

[3] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMatematicheskii Sbornik vol 64 no 2 p 427 1989

[4] V A Bailov R K Gazizov and N H Ibragmov ldquoPertubationmethods in group analysisrdquo Journal of Mathematical Sciencesvol 55 no 1 pp 1450ndash1485 1991

[5] N H Ibragimov and V F Kovalev Approximate and Renor-mgroup Symmetries Higher Education Press Beijing China2009

[6] V A Baikov and N H Ibragimov ldquoContinuation of approxi-mate transformation groups via multiple time scales methodrdquoNonlinear Dynamics vol 22 no 1 pp 3ndash13 2000

[7] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations vol 168 ofApplied Mathematical Sciences Springer New York NY USA2010

[9] A H Kara F M Mahomed and G Unal ldquoApproximate sym-metries and conservation laws with applicationsrdquo InternationalJournal ofTheoretical Physics vol 38 no 9 pp 2389ndash2399 1999

[10] A G Johnpillai A H Kara and FMMahomed ldquoApproximateNoether-type symmetries and conservation laws via partial

Advances in Mathematical Physics 11

Lagrangians for PDEs with a small parameterrdquo Journal of Com-putational and AppliedMathematics vol 223 no 1 pp 508ndash5182009

[11] J-R Su S-L Zhang and J-N Li ldquoApproximate noether-typesymmetries and conservation laws via partial lagrangians fornonlinear wave equation with dampingrdquo Communications inTheoretical Physics vol 53 no 1 pp 37ndash42 2010

[12] E S Benilov and B A Malomed ldquoOn the spectral singularityproblem in the perturbation theory for systems close to theKdVequationrdquo Journal of Physics A Mathematical and General vol21 no 8 pp L439ndashL442 1988

[13] M A Allen and G Rowlands ldquoA solitary-wave solution to aperturbedKdV equationrdquo Journal of Plasma Physics vol 64 no4 pp 475ndash480 2000

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Advances in Mathematical Physics

11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905) 1199062 + 119895 (119905 119909)

(84)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909)119891(119905 119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 and 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetric operators

119884 = 120576119892 (119905) 120597120597119906 (85)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12120576119906119909119892 (119905)

1198792 = 119891 (119905 119909) + 120576 [minus12119892119905 (119905) 119906 + 119895 (119905 119909)]minus [120576119892 (119905)] [31199062119909 minus 1

2119906119905] + 119863119909 [120576119892 (119905)] 119906119909119909(86)

Second Case 120572 = 2120573 and 120573 = 212057212058510 = 012058520 = 01205780 = ℎ (119905) 12058511 = 012058521 = 01205781 = 119892 (119905) 11986110 = 119890 (119905 119909) 11986120 = minusℎ119905 (119905) 1199062 + 119891 (119905 119909) 11986111 = 119894 (119905 119909) 11986121 = minus119892119905 (119905)2 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)

(87)

where 119892 and ℎ are arbitrary function for 119905 and 119890(119905 119909) 119891(119905119909) 119894(119905 119909) 119895(119905 119909) are differential functions and 119890119905(119905 119909) +119891119909(119905 119909) = 0 119894119905(119905 119909) + 119895119909(119905 119909) = 0Thus the equation has the following approximate Noether

symmetry operators

119884 = ℎ (119905) 120597120597119906 + 120576119892 (119905) 120597

120597119906 (88)

So the conservation vectors are

1198791 = 119890 (119905 119909) + 120576119894 (119905 119909) + 12119906119909 [ℎ (119905) + 120576119892 (119905)]

1198792 = 119891 (119905 119909) + [minus12ℎ119905 (119905) 119906]+ 120576 [minus12119892119905 (119905) 119906 minus 3120572ℎ (119905) + 119895 (119905 119909)]minus [120576119892 (119905) + ℎ (119905)] [31199062119909 minus 1

2119906119905]+ 119863119909 [120576119892 (119905) + ℎ (119905)] 119906119909119909

(89)

Third Case 120572 = 2120573 and 120573 = 2120572 the conservation laws are thesame as the first case

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Thisworkwas supported in part by theNatural Science Foun-dation of Inner Mongolia of China under grant 2016MS0116

References

[1] A Wazwaz Partial Differential Equations and Solitary WavesTheory Higher Education Press Beijing China 2009

[2] Y S Kivshar and B A Malomed ldquoDynamics of solitons innearly integrable systemsrdquo Reviews of Modern Physics vol 61no 4 pp 763ndash915 1989

[3] V A Baıkov R K Gazizov and N H Ibragimov ldquoApproximatesymmetriesrdquoMatematicheskii Sbornik vol 64 no 2 p 427 1989

[4] V A Bailov R K Gazizov and N H Ibragmov ldquoPertubationmethods in group analysisrdquo Journal of Mathematical Sciencesvol 55 no 1 pp 1450ndash1485 1991

[5] N H Ibragimov and V F Kovalev Approximate and Renor-mgroup Symmetries Higher Education Press Beijing China2009

[6] V A Baikov and N H Ibragimov ldquoContinuation of approxi-mate transformation groups via multiple time scales methodrdquoNonlinear Dynamics vol 22 no 1 pp 3ndash13 2000

[7] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1986

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations vol 168 ofApplied Mathematical Sciences Springer New York NY USA2010

[9] A H Kara F M Mahomed and G Unal ldquoApproximate sym-metries and conservation laws with applicationsrdquo InternationalJournal ofTheoretical Physics vol 38 no 9 pp 2389ndash2399 1999

[10] A G Johnpillai A H Kara and FMMahomed ldquoApproximateNoether-type symmetries and conservation laws via partial

Advances in Mathematical Physics 11

Lagrangians for PDEs with a small parameterrdquo Journal of Com-putational and AppliedMathematics vol 223 no 1 pp 508ndash5182009

[11] J-R Su S-L Zhang and J-N Li ldquoApproximate noether-typesymmetries and conservation laws via partial lagrangians fornonlinear wave equation with dampingrdquo Communications inTheoretical Physics vol 53 no 1 pp 37ndash42 2010

[12] E S Benilov and B A Malomed ldquoOn the spectral singularityproblem in the perturbation theory for systems close to theKdVequationrdquo Journal of Physics A Mathematical and General vol21 no 8 pp L439ndashL442 1988

[13] M A Allen and G Rowlands ldquoA solitary-wave solution to aperturbedKdV equationrdquo Journal of Plasma Physics vol 64 no4 pp 475ndash480 2000

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Advances in Mathematical Physics 11

Lagrangians for PDEs with a small parameterrdquo Journal of Com-putational and AppliedMathematics vol 223 no 1 pp 508ndash5182009

[11] J-R Su S-L Zhang and J-N Li ldquoApproximate noether-typesymmetries and conservation laws via partial lagrangians fornonlinear wave equation with dampingrdquo Communications inTheoretical Physics vol 53 no 1 pp 37ndash42 2010

[12] E S Benilov and B A Malomed ldquoOn the spectral singularityproblem in the perturbation theory for systems close to theKdVequationrdquo Journal of Physics A Mathematical and General vol21 no 8 pp L439ndashL442 1988

[13] M A Allen and G Rowlands ldquoA solitary-wave solution to aperturbedKdV equationrdquo Journal of Plasma Physics vol 64 no4 pp 475ndash480 2000

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom