Families of particular solutions to multidimensional partial differential equations

Families of particular solutions to multidimensional partial differential equationsA. I. Zenchuk Citation: Journal of Mathematical Physics 42, 5472 (2001); doi: 10.1063/1.1398337 View online: http://dx.doi.org/10.1063/1.1398337 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/42/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in One-dimensional coupled Burgers’ equation and its numerical solution by an implicit logarithmic finite-differencemethod AIP Advances 4, 037119 (2014); 10.1063/1.4869637 The solutions of partial differential equations with variable coefficient by Sumudu Transform Method AIP Conf. Proc. 1493, 91 (2012); 10.1063/1.4765475 A note on the numerical solution of fractional Schrödinger differential equations AIP Conf. Proc. 1470, 92 (2012); 10.1063/1.4747647 Multidimensional partial differential equation systems: Generating new systems via conservation laws, potentials,gauges, subsystems J. Math. Phys. 51, 103521 (2010); 10.1063/1.3496380 Upper and lower solutions for periodic problems: first order difference vs first order differential equations AIP Conf. Proc. 835, 30 (2006); 10.1063/1.2205034

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Families of particular solutions to multidimensional partialdifferential equations

A. I. Zenchuka)

Department of Mathematics, The University of Arizona, Tucson, Arizona 85721

~Received 22 September 2000; accepted for publication 11 July 2001!

Dressing methods are known as productive tools for construction of the particularsolutions to the big class of nonlinear partial differential equations~PDEs! whichare integrable by the inverse scattering technique. Recently the modification of thedressing method based on the system of algebraic equations has been suggestedwhich allows us to find the families of particular solutions to certain types ofnonintegrable~in classical sense! nonlinear PDEs. This modification representsPDEs as closure reductions of an appropriate differential-difference system. In thisarticle we study the dressing procedure in more detail. Particularly, we considerdifferent families of particular solutions available through the dressing methodbased on the algebraic system. We give two examples of the differential-differencesystems and related PDEs and point to other possible generalizations of the dress-ing method. ©2001 American Institute of Physics.@DOI: 10.1063/1.1398337#

I. INTRODUCTION

Dressing method has been developed as a special technique for solving the nonlinear inte-grable partial differential equations~PDEs!.1–4 Many modifications of the dressing method havebeen found.5–8An important field of application of the dressing methods is the investigation of the(n11)-dimensional (n.2) integrable systems.8–12

Recently13 the algebraic system ofN(c) equations

L~cm![cm2(i 51

N~c!

c i (n51

N~r !

cinRnm5hm , m51,...,N~c! or

~1!L~c![c2cCR5h, c5@c1¯cN~c!#, h5@h1¯hN~c!#

had been suggested as an auxiliary equation for construction of the particular solutions to the PDEof a certain types. In Eq.~1! we use constantN(c)3N(r ) ~N(c)ÞN(r ) in general! matrix C5$cin% to provide the possibility to use rectangular~not only square! N(r )3N(c) matrix R5$Rnm%. Although the algebraic system can be taken as a starting point for the following discus-sion, it is important to emphasize its relation with the classical dressing method. For simplicity werefer to the]-problem,5,6 since the above system of algebraic equations can be considered as thediscrete version of this problem. In fact, the]-problem represents the relation between solutions ofthe nonlinear PDE and solutions of the following linear integral equation:14

c~l!2E c~n!c~n,m!R~m,l!dm∧dmdn∧dn5h~l!, ~2!

whereR is a kernel of the integral operator andh is a normalization function. Parametersn, m, lare complex. The additional parametersx5(x1 ,...,xM) are introduced in the kernelR by theformulas

a!Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 42, NUMBER 11 NOVEMBER 2001

54720022-2488/2001/42(11)/5472/21/$18.00 © 2001 American Institute of Physics


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] iR~m,l!5 (k51

Ni

f k~ i !~m!gk

~ i !~l!, i 51,...,M , ~3!

which are compatible provided

] i (k51

Nj

f k~ j !~m!gk

~ j !~l!5] j (k51

Ni

f k~ i !~m!gk

~ i !~l!, i , j 51,...,M . ~4!

The link to the particular system of nonlinear PDEs is produced by the special structure of thefunctions f k

( j )(m) andgk( j )(l). Parametersx are independent variables of this system.5,6 Since the

conditions~4! should be satisfied for all admitted values of the parametersm andl, there is nofunctional relations between functionsf k

( j )(m) andgk( j )(l). This conclusion makes restrictions on

the admitted form of the functionsf k( j )(m) and gk

( j )(l). For this reason integral equation~2! isrelated only with completely integrable systems of nonlinear PDEs.

The situation becomes different if one considers the solution of Eq.~2! on the set of discretepointsn i ,mn ,lm ( i ,m51,...,N(c),n51,...,N(r )) on the complex planes of parametersn, m, l andintroduces matricesC andR with the entries

cin[c~n i ,mn!, Rnm[R~mn ,lm!.

Then one gets the algebraic system~1! with the following discrete version of the equations~3! and~4!

] iRnm5 (k51

Ni

f nk~ i !gkm

~ i ! , n51,...,N~r !, m51,...,N~c!, i 51,...,M , or

~5!

] iR5 (k51

Ni

f~ ik !g~ki !, f~ ik !5@ f 1k~ i !¯ f N~r !k

~ i !#T, g~ki !5@gk1

~ i !¯gkN~c!

~ i !#,

] i (k51

Nj

f nk~ j !gkm

~ j !5] j (k51

Ni

f nk~ i !gkm

~ i ! , i , j 51,...,M , or

~6!

] i (k51

Nj

f~ jk !g~k j !5] j (k51

Ni

f~ ik !g~ki !

where

f nk~ j !5 f k

~ j !~mn!, gkm~ j !5gk

~ j !~lm!.

It turned out that the discrete version has a set of properties which makes it very adaptable inapplications to different systems of nonlinear PDEs. The basic reason for this is the absence ofcomplex continuous parameters in the equations~6!, which become bilinear differential equationson the functionsf nk

( j ) andgkm( j ) which are functions of parametersx only. In general the associated

systems of nonlinear PDEs are not integrable, but integrable systems are situated among them.One can recognize that the equations~1! and ~2! can be combined into the following one:

cm~l!2E dm∧dmdn∧dn(k51

N~c!

ck~n! (n51

N~r !

ckn~n,m!Rnm~m,l;x!5hm~l!, m51,...,N~c!,

~7!

with

5473J. Math. Phys., Vol. 42, No. 11, November 2001 Families of particular solutions


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] iRnm~m,l!5 (k51

Ni

f nk~ i !~m!gkm

~ i ! ~l!. ~8!

Equation~7! will not be discussed in this article.For our purpose we require the following properties of the matrixR:13

~1! The nonhomogeneous equation~1! has the unique solution,c5@c1¯cN(c)#.~2! Thex dependence@x5(x1 ,...,xM), M is dimension ofx-space# is introduced by the formulas

~5!.~3! The equations~5! are compatible, i.e., the set of conditions~6! is held which is the system of

PDEs imposed on the functionsf nk( i ) andgkm

( i ) .~4! The set of self-consistent constraints is imposed on the functionsg(ki) in the form of the

system of PDEs

(ik

Likm~g! g~ki !50, m51,...,P ~9!

~whereL ikm(g) are scalar linear differential operators andg(ki) is 13N(c) row!. These constraints

should be consistent with the compatibility conditions~6!. We will use operatorsL ikm(g) with

constant coefficients for simplicity.

The consequence of the first requirement is the fact that homogeneous algebraic system with thesame matrixR has only the trivial solution, i.e.,

L~f!50⇔f[0. ~10!

Equations~1! and~5! produce the following system of algebraic equations~with the same matrixR and different right-hand sides! for the functions] jc, ] j

nc and)s51M ]s

nsc:

~11!

~12!

Cnp5

p!

n! ~n2p!!,

LS )s51

M

]snsc D 5hn1 ...nM

, ~13!

hn1 ...nM5]mhn1 ...~nm21!...nM

1 (k51

Nj S (p150

n1

¯ (pm50

nm21

¯ (pM50

nM

~21!( i 51M pi

3S )s51,sÞm

M

Cns

ps]sns2psD Cnm21

pm ]mnm2pm21Up1 ,...,pM

~ jk ! g~ jk !D ,

where

Un1 ...nM

~ ik ! 5 (n51

N~c!

cn (m51

N~r ! S )s51

M

]snsD cnmf mk

~ i ! [cCS )s51

M

]snsD f~ ik !,

i 51,...,M , k51,...,Ni . ~14!

5474 J. Math. Phys., Vol. 42, No. 11, November 2001 A. I. Zenchuk


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Using these algebraic systems together with the original system~1! and applying the superpositionprinciple for linear nonhomogeneous algebraic systems, one can provideM (L) differential opera-tors M j such thatL(M jc)50 or in accordance with Eq.~10!

M jc50, j 51,...,M ~L !, M ~L !>1. ~15!

One considers Eqs.~15! as the system of linear PDEs on the functionc. In addition we require theindependence of the operatorsM k , i.e., there is no nonzero set of differential operatorsD j , j

51,...,M (L), such that( j 51M (L)

D j M j50.The functionsUn1 ,...,nM

( ik) defined by Eq.~14! are solutions of the appropriate system of

differential-difference equations, which can be derived from the linear system~15! ~where eachequation is 13N(c) row! by multiplying it by theN(c)31 vector

vn1...nM

ik [CS )s51

M

]snsD f~ ik !. ~16!

Let U be the set of all matrices~14!. Then the nonlinear system is the system of scalar equationsof the general form

M jcvn1,...,nM

ik [Mn1 ,...,nM

ik j ~U!50, j 51,...,M ~L !, i 51,...,M , k51,...,Ni . ~17!

By construction the total number of discrete parameters equals the number of continuous param-etersM in the system~17!.

The system~17! represents the general system of differential-difference equations, which isrelated with the algebraic system~1!. The system~17! is the system of pure PDEs if there arenumbersM ( ik js)>0 (s51,...,M ) such that the~sub!set of the equations~17! with subscripts

~n1 ,...,nM !<~M ~ ik j 1!,...,M ~ ik jM !!, j 51,...,M ~L !, i 51,...,M , k51,...,Ni

~each entry on the left-hand side does not exceed the correspondent entry on the right-hand side!represents the complete system of PDEs.

Otherwise, the system of pure PDEs can be derived from the system~17! by imposing thereductions which introduce the relation among different discrete parametersnj and reduce theirnumber fromM to M,M in such a way that the~sub!system of the system~17! forms thecomplete system of pure nonlinear PDEs, i.e., the numbersM ( ik js)>0 (s51,...,M ) should existsuch that the~sub!set of the equations~17! with subscripts

~n1 ,...,nM !<~M ~ ik j 1!,...,M ~ ik jM !!

forms the complete system of pure PDEs. To clarify the general form of these reductions, note thateach discrete parameternj is associated with the derivative] j f

( ik). It means that one needs toimpose the relations among thederivativesof the functionsf( ik). Then one gets the relationsamong the functionsUn1 ...nM

( ik) with fixed superscripts~ik! and different subscripts (n1 ,...,nM) and,

consequently, the relations among the discrete parameters. It means that the reductions of ourinterest have the general form

(in

L ikm~ f ! f~ ik !50, m51,...,P, ~18!

whereL ikm( f ) are the scalar linear differential operators~with constant coefficients for simplicity!.

Both systems~9! and ~18! should be self-consistent and compatible with the system~6!. It is



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important to emphasize that the reductions which have the form ofalgebraic relations amongdifferent functionsf( ik) do not reduce the number of discrete parameters in the system.

The significant difference between integrable PDEs and ones considered in this article is thatthe integrable nonlinear systems have operator representation as the compatibility condition of theappropriate overdetermined linear system of equations with variable coefficients~zero curvaturerepresentation!. The nonlinear PDEs under consideration are related with the systems of linearequations~15! as well, but correspondent operator representation for them has not been found yet.

In the next section~Sec. II! we develop the general approach for construction of the particularsolutions for the systems of nonlinear PDEs, associated with the algebraic system~1!. In Sec. IIIwe consider the examples of the nonintegrable generalizations of the modified Kadomtsev–Petviashvili equation~mKP! with several families of particular solutions~some of them are givenin Ref. 13!. In Sec. IV we apply new dressing method to find the possible relations amongdifferent integrable hierarchies, for instance, between two different mKP hierarchies. Some gen-eralizations of the dressing method based on the algebraic systems are discussed in Sec. V. We willmention about the algebraic system~1! with non-unit right-hand side, which serves, for example,the generalizations of the Kadomtsev–Petviashvili equation~KP! ~see the Appendix!. Conclusionsare given in Sec. VI.

II. FAMILIES OF PARTICULAR SOLUTIONS

Next we discuss the available families of particular solutions to the nonlinear PDE relatedwith the algebraic system~1!. The main problem in construction of particular solutions is to satisfythe compatibility condition~6! together with constrains~9! and ~18! for all pairs of variables(xi ,xj ),x. In general, forM-dimensional space one has to satisfy (M21)M /2 conditions eachrepresented byN(r )3N(c) equations. But under assumption that all products of the functionsf nk

( i )gkm( i ) in ~5! depend on complete set of variables,x, the number of independent conditions

becomes equal toM21, i.e., the system~6! can be replaced, for instance, with the following one:

]1(k51

Nj

f~ jk !g~k j !5] j (k51

N1

f~1k!g~k1!, j 52,...,M . ~19!

We consider only such matricesR which give rise to the solutionc5@c1 ,...,cN(c)# of the alge-braic system~1! composed of linearly independent functionscm ,m51,...,N(c):

functions cm ~m51,...,N~c!! in the solution c

5@c1 ,...,cN~c!# of the system~1! are linearly independent. ~20!

The basic factor which defines the variety of the solutions is the allowed dimensionsN(r ) andN(c)

of the matrixR in the algebraic system~1!. Although the form of nonlinear PDEs does not dependon them, the admitted values of these dimensions are related essentially with the particular situ-ation and are defined by both the equations~6! and the relations~9! and~18!. In general, differentfamilies of particular solutions can be characterized by the four numbers,Nmin

(r) , Nmax(r) , Nmin

(c) , Nmax(c) ,

which represent the minimum and maximum possible values for the parametersN(r ) and N(c),respectively, so thatNmin

(r) <N(r)<Nmax(r) andNmin

(c) <N(c)<Nmax(c) . For our convenience we combine all

functionsf( ik) andg(ki) related with each family in the manifoldsFNmin(r) N

max(r)

iandGN

min(c) N

max(c)

i, respec-

tively. Note that condition~20! requires that parametersNmin(r) , Nmax

(r) , Nmin(c) , Nmax

(c) do not depend oni, i 51,...,M21. We collect these two sets of manifolds in two general manifoldsF5ø

iFN

min(r) N

max(r)

i

andG5øi

GNmin(c) N

max(c)

i. Functions from these manifolds we callf- andg-functions, respectively.

We consider three steps of the investigation of the particular solutions, which can be appliedto any nonlinear PDE, generated by the algebraic system~1!. More detailed investigation can be



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specifically performed for each particular system. Subscriptsn andm run the values from 1 toN(r )

and from 1 toN(c), respectively, in all formulas below, unless otherwise specified.Step 1.(Criterion for application of the dressing method.)First of all, one needs to check if

the compatibility conditions~19! can be satisfied at least forN(r )5N(c)51. If this is possible, thenthe related system of nonlinear PDEs can be treated by the dressing method and one has at leasta family of particular solutions for whichf- andg-functions are collected in the manifoldsF1,1

i andG1,1

i .After that we have to study the possibility to satisfy Eqs.~19! with N(r ).1 and/orN(c).1. In

general the system~19! is nonlinear and it is difficult to solve it. But one can separate the familiesof particular solutions for this system which can be received by ‘‘splitting’’ the equations~19! intothe set of linear differential~or algebraic! equations on thef- and g-functions ~or their Fourieramplitudes!. We assume that allf- andg-functions depend on the whole set of variablesx. Due tothe reductions~9! and ~18! the equations~19! may involve the partial derivatives off- andg-functions with respect to different sets of variablesx1 andx2 , respectively (x1 ,x2#x). Step 2regards the situation when at least one of the setsx1 or x2 coincides withx.

Step 2.Let us impose the set of reductions~9! and~18!, assume thatx1[x ~the casex2[x canbe considered in the analogous way, see Sec. III!. Then the system~19! can be represented as thesum of the products:

(k51

Pi

Fnki Gkm

i 50, i 51,...,M21,

Fnki 5Fk

i ~ f np~ j ! , j 51,...,M , p51,...,Nj !, ~21!

Gkmi 5Gk

i ~gpm~ j ! , j 51,...,M , p51,...,Nj !

where we denoteFnki and Gkm

i the linear combination of the functionsf np( j ) and gpm

( j ) ( j51,...,M ,p51,...,Nj ), respectively, and their derivatives ini th equation of the system~19!. Let usput Nmin

(r) 51. To findNmin(c) andNmax

(c) , let us fixn51 in the system~21!:

(k51

Pi

F1ki Gkm

i 50, i 51,...,M21, m51,...,N~c!, ~22!

and consider the last system as a system of equations on the functionsF1k( i ) .

Let us introduce several definitions. We call the set of functions$Fnki , i is fixed, k

51,...,Gi(g)% independent if for any permitted value of parametern there is an appropriate set of

functions f n,$ f nk( i ) ,i 51,...,M ,k51,...,Ni% with the same length such that one can establish the

uniquely invertible map$Fnki %↔ f n . Analogously, we call the set of functions$Gkm

i , i is fixed,k51,...,Di

( f )% linearly independent, if for any admitted value of parameterm there is an appropri-ate set of functionsgm,$gkm

( i ) ,i 51,...,M ,k51,...,Ni% with the same length such that one canestablish the uniquely invertible map$Gkm

i %↔gm . Let Di(g) be the minimum number of indepen-

dent functionsFki for each fixedi andD (g)5min(Di

(g)). One can always reenumerate functionsFki

in such a way that functionsFki are independent fork51,...,Di

(g) . We impose two requirements:

~1! D (g)>M21, otherwise the succeeding consideration in the scopes ofStep 2leads to thenonlinear equations ong-functions.

~2! All functions Fki with k51,...,D (g) and i 51,...,M21 are independent, i.e., one has (M

21)D (g) independent functions altogether.

Then for any numberD (g)<D (g) the system~22! with m51,...,D (g) is the linear system on thefunctionsFk

i ~or on f-functions!, which can be resolved provided an appropriate choice of thearbitraryg-functions is made. In this way we have fixedNmin

(c) 5Nmax(c) 5D(g) in the system~21!. Let



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us denote D i( f ) the number of independent solutions of the equations~22! with D ( f )

5min(Di(f )). ThenNmin

(r) 51 andNmax(r) 5D(f ). So, f- andg-functions are combined in the manifolds

F1D( f )i

andGD(g)D(g)i

~D (g)51,...,D (g) and D ( f ) depends onD (g)!. The matrixR in the algebraic

system ~1! is represented by its elementsRnm5] i21Sk51

Ni f nk( i )gkm

( i ) with some fixedi and f( ik)

PF1D( f )i

,g( ik)PGD(g)D(g)i

. We emphasize that allg-functions are arbitrary functions of the set ofvariablesx in the above formulas.

In the previous consideration we separated the part of the system~22! with m<D (g). Now letus consider the part of the system~22! with m.D (g). If we are able to provideN equations in thissystem, which are compatible with the first part of the system and appropriate functionsgkm

( i ) arelinearly independent with functionsgkm

( i ) , m<D (g), then the parameterNmax(c) will become equal to

D (g)1N. In fact, let us consider the subsystem of the system~22! with m51,...,D (g) as analgebraic system, resolvable for the functionsFk

i ~i 51,...,M21, k51,...,Pi!. Let n run values from1 to D* ( f )<D ( f ). Substitute the solution of this subsystem into the equations of the system~21!

with m.D (g) and consider the resultant system as the set of identities for alln51,...,D* ( f ). Itleads to either the linear or nonlinear equations ong-functions. IfN is the number of solutions ofthis system linearly independent withg-functions fixed before, thenNmax

(c) 5D(g)1N. So,g-functions belong to the manifoldsG

D(g)(D(g)1N)

i, with D (g)51,...,D (g), where N depends on

D* ( f ). f-functions belong to the manifoldsF1D* ( f )i

.Even if one of the setsx1 or x2 coincides withx, it is not necessary that the system~19! be

treated by theStep 2. It happens, for instance, ifD (g),M21. The solutions provided by the nextstep are available always if only thecriterion is satisfied.

Step 3. Since the constrains~9! and ~18! are given by the linear differential equations withconstant coefficients, one can representf- andg-functions in terms of Fourier integrals

f nk~ i !5E ank

~ i !~kn!eknx11 knx1 dkn , ~23!

gkm~ i ! 5E bkm

~ i ! ~vm!evmx21vmx2 dvm , ~24!

kn5~kn1 ,...,kn dim~x1!!, kn5~kn dim~x1!11 ,...,knM!, ~25!

vm5~vm1 ,...,vm dim~x2!!, vm5~vm dim~x2!11 ,...,vmM!,

dvm5dvm1 ...dvm dim~x2! , dkn5dkn1 ...dkn dim~x1! ,

xiù xi5B, xiø xi5x,

wherekn(kn) andvn(vn) represent the dispersion relation for the linear differential equations~9!and~18!. In order to satisfy the compatibility conditions~21! in the nonintegrable case, one needsto provide the possibility to establish the relations between spectral parameterskn and vn . Forthis purpose one needs to replace at least one of the equations~23! or ~24! by the finite Fourierseries. Namely, the two following representations forf- andg-functions can be used~for the sakeof simplicity we take the single Fourier harmonic instead of the finite series!:

f nk~ i !5ank

~ i !eknx11 knx1, gkm~ i ! 5E bkm

~ i ! ~vm!evmx21vmx2dvm ~26!

and



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f nk~ i !5E ank

~ i !~kn!eknx11 knx1 dkn , gkm~ i ! 5bkm

~ i ! evmx21vmx2. ~27!

The examination of each of these representations is equivalent, so that we study only the first one.The compatibility conditions~21! become the system of algebraic equations on the parameters

ank( i ) , kn j,bkm

( i ) , wm j :

(k51

Pi

Fki ~an ,kn!Gk

i ~bm ,vm!50, i 51,...,M21, ~28!

an5~ank~ i ! , i 51,...,M , k51,...,Ni !, ~29!

bn5~bkn~ i ! , i 51,...,M , k51,...,Ni !, ~30!

with knÞkm and vnÞvm if nÞm in order to satisfy the requirement~20!. This system admitsfollowing families of solutions:

~1! Let us keep independent continuous parametersvmk with different k @k51,...,dim(x2)#and m in the representation~26!. AssumeNmin

(r) 5Nmin(c) 51 and find out the appropriate values for

Nmax(r) andNmax

(c) . First of all let us fixn51 in the above system, so that it can be written in the form

(k51

Pi

Fki ~a1 ,k1!Gk

i ~bm ,vm!50, i 51,...,M21. ~31!

Since parametersvmk are independent, the above equations represent the relations among param-etersbkm

( i ) . So, one hasM21 relations among these parameters for each particular value of theparameterm. This means that if the system~31! is consistent for some fixedm, then it is consistentfor any m, i.e., Nmax

(c) 5` and theg-functions belong to the manifoldG1`( i ) . Otherwise the system

~31! is inconsistent. Assuming the consistence of the system~31!, let us defineNmax(r) . Suppose that

Nmax(r) .1 and consider two equations~28! with n51 andn5p<N(r ). Subtract one equation from

another to get

(k51

Pi

„Fki ~a1 ,k1!2Fk

i ~ap ,kp!…Gki ~bm ,vm!50, i 51,...,M21. ~32!

Since this equation should be satisfied for all admitted values of the parameterm, one gets the setof algebraic equations on parametersap

( i ) andkp :

„Fki ~a1 ,k1!2Fk

i ~ap ,kp!…50, k51,...,Pi , i 51,...,M21. ~33!

If set (ap ,kp),p51,...,n11, represents solutions of this system with different spectral parameters~i.e.,knÞkp if nÞp for n,p51,...,n11!, thenNmax

(r) 5n11 andf-functions belong to the manifoldsF1(n11)

( i ) .We assumed that parametersvmk with different k are independent in the formulas~31!–~33!,

which is not always possible@see Sec. III A, Eqs.~70!–~72!#. The following family of solutions isnot based on this assumption.

~2! For this family of solutions we impose the additional relations among parametersvmk withdifferent k. These relations can be established in the following way. Let us consider the system~31!. For each particularm we selectM<M21 equations of these systems to impose the relationsamongmb parametersbkm

( i ) andm<dim(x2) parametersvmk(mb1m5M ), i.e., split the completesystem~31! into two subsystems,



129.105.215.146 On: Sun, 21 Dec 2014 15:38:36

(k51

Pi

Fki ~a1 ,k1!Gk

i ~bm ,vm!50, i 51,...,M , ~34!

(k51

Pi

Fki ~a1 ,k1!Gk

i ~bm ,vm!50, i 5M11,...,M21, ~35!

and consider the first system as an algebraic system of equations formb parametersbkm( i ) and m

parametersvm j . This means thatNmin(c) 51. If m,dim(x2) and the system~34! is resolvable for

some particularm, then it is resolvable for arbitrarym with „dim(x2)2m… arbitrary parametersvmk for each particularm, i.e.,Nmax

(c) is infinity, g-functions belong to the manifoldsG1`( i ) , and the

integral in ~26! becomes„dim(x2)2m…-dimensional. Ifm5dim(x2), then there are no arbitraryparametersvmk , integration disappears from the Eq.~26!, and the maximum dimensionNmax

(c) isdefined by the numberm0 of the solutions of the system~32! with different spectral parametersvm : vmÞvp if mÞp for all m,p51,...,m0 . Consequentlyg-functions belong to the manifoldG1m0

( i ) . Now we have to satisfy the system~35!. First of all one needs to substitute the established

relations among parametersbm( i ) andvm into the system~35! to end up with the system of the form

(k51

Pi

Fki ~a1 ,k1!Gk

i ~bm ,vm!50, i 5M11,...,M21. ~36!

Considering this system as a set of identities for all admittedm one gets the following system ofequations for parametersa1 andk1 :

Fki ~a1 ,k1!50, k51,...,Pi , i 5M11,...,M21.

If this system is consistent, then the imposed relations amongbkm( j ) andvmi are allowed at least for

Nmin(r) 51.

To find Nmax(r) one has to assume thatNmax

(r) .1 in ~28!, substitute all found relations amongparameters into the equations of the system~28! with 1,n<Nmax

(r) , and consider them as identitiesfor all m. One gets the system of algebraic equations on the parametersan andkn . The number ofits solutionsn with differentkn ~knÞkp if nÞp, n,p51,...n11! defines the maximum dimensionNmax

(r) : Nmax(r) 5n11. So,f-functions belong to the manifoldsF1,n11

( i ) .~3! To establish the relations among parametersbkm

( i ) andvmk one can take the subsystem ofn0

equations out of the system~28! with n51,...,n0 . The algorithm is exactly the same. One canestablishM relations amongmb parametersbkm

( i ) and m<dim(x2) parametersvmk for each par-ticular m (mb1m5M ). If the remainingn0(M21)2M equations are compatible, one gets therelations among parametersank

( i ) and knk (n51,...,n0), which mean thatNmin(r) 5n0. To find Nmax

(r)

one needs to assume thatNmax(r) .n0, substitute all found relations into the equation~28! with n0

,n<Nmax(r) , consider them as identities for allm, and solve the appropriate system of algebraic

equations on the parametersank( i ) andknk . If the set (ap ,kp), p51,...,n1n0 , represents solutions

of this system with different spectral parameterskp ~i.e., knÞkp if nÞp and n,p51,...,n1n0!,thenNmax

(r) 5n1n0 and f-functions are in the manifoldsFn0 ,n1n0

( i ) . g-functions belong to the mani-

folds G1`( i ) or G1m0

( i ) if m5dim(x2). In the last casem0 is the number of solutions (bm ,vm) to the

system~28! ~n51,...,n0 , i 51,...,M ! with different vm .

III. „3¿1…-DIMENSIONAL GENERALIZATION OF mKP

In this section we use the algorithm described above to derive the (311)-dimensional non-integrable generalization of mKP. We use (311)-dimensional equations to show how this algo-rithm works in multidimension, where the classical integrability theory is applicable only tospecial types of PDEs. There is no formal restriction on the dimension of PDEs, which can be



129.105.215.146 On: Sun, 21 Dec 2014 15:38:36

treated by the method considered in this article. One can construct (211)-dimensional noninte-grable generalizations of mKP which will involve different types of nonlinearity. Although thegeneral nonlinear equations@see Eqs.~47! and ~48!# are rather cumbersome, they admit a set ofreductions which reduce the general system into the simpler equations like Eqs.~61!–~63!,13

which may be regarded as the simplest generalizations of mKP~64!.To start with, let us takeM54 ~Ref. 13! and normalizationhm51, m51,...,N(c), in the

system~1!. Introducex-dependence by the set of equations~5!

]1R5f~11!g~1!, ]2R5f~21!g~2!, ]3R5f~31!g~1!1f~32!]1g~1!,

]4R5f~41!g~2!1f~42!]2g~2!1f~43!]22g~2!, ~37!

g~ i !5@g1~ i !¯gN~c!

~ i !#, f~ ik !5@ f 1k

~ i !¯ f N~r !k

~ i !#T.

To construct the auxiliary linear system, let us write down the set of linear algebraic systems~11!–~13!, which is generated by the original system~1! and Eqs.~37!:

L~]1c!5U0,0,0,0~11! g~1!, ~38!

L~]12c!5~2]1U0,0,0,0

~11! 2U1,0,0,0~11! !g~1!1U0,0,0,0

~11! ]1g~1!, ~39!

L~]2c!5U0,0,0,0~21! g~2!, ~40!

L~]22c!5~2]2U0,0,0,0

~21! 2U0,1,0,0~21! !g~2!1U0,0,0,0

~21! ]2g~2!, ~41!

L~]23c!5~3]2

2U0,0,0,0~21! 23]2U0,1,0,0

~21! 1U0,2,0,0~21! !g~2!

1~3]2U0,0,0,0~21! 2U0,1,0,0

~21! !]2g~2!1U0,0,0,0~21! ]2

2g~2!, ~42!

L~]3c!5U0,0,0,0~31! g~1!1U0,0,0,0

~32! ]1g~1!, ~43!

L~]4c!5U0,0,0,0~41! g~2!1U0,0,0,0

~42! ]2g~2!1U0,0,0,0~43! ]2

2g~2!, ~44!

Un1 ,n2 ,n3 ,n4

~ ik ! 5 (m51

N~r !

cm(n51

N~c!

cmn]1n1]2

n2]3n3]4

n4f nk~ i ![cC]1

n1]2n2]3

n3]4n4f~ ik !, ~45!

c5@c1 ...cN~c!#.

Then the linear system~15! has the form

M1c[]3c1V1]12c1V2]1c50,

~46!M2c[]4c1W1]2

3c1W2]22c1W3]2c50

with

V152U0,0,0,0~32! /U0,0,0,0

~11! , V25„2U0,0,0,0~31! 2V1~2]1U0,0,0,0

~11! 2U1,0,0,0~11! !…/U0,0,0,0

~11! ,

W152U0,0,0,0~43! /U0,0,0,0

~21! , W25„2U0,0,0,0~42! 2W1~3]2U0,0,0,0

~21! 2U0,1,0,0~21! !…/U0,0,0,0

~21! ,

W35„2U0,0,0,0~41! 2W1~3]2

2U0,0,0,0~21! 23]2U0,1,0,0

~21! 1U0,2,0,0~21! !2W2~2]2U0,0,0,0

~21! 2U0,1,0,0~21! !…/U0,0,0,0

~21! .

In fact, one can check directly thatL(Mic)50, i 51,2. Note that each of the equations~46! is amatrix 13N(c) equation with scalar coefficientsVi and Wi . To get a nonlinear system let us



129.105.215.146 On: Sun, 21 Dec 2014 15:38:36

multiply them by theN(c)31 vector functionsC]1n1]2

n2]3n3]4

n4f( i j ) and use the definition of poten-tials Un1,n2,n3,n4

( ik) given by~45!. One result is the following differential-difference system with fourcontinuous and four discrete parameters:

]3Un1 ,n2 ,n3 ,n4

~ i j ! 2Un1 ,n2 ,n311,n4

~ i j ! 1V1~]12Un1 ,n2 ,n3 ,n4

~ i j ! 22]1Un111,n2 ,n3 ,n4

~ i j ! 1Un112,n2 ,n3 ,n4

~ i j ! !

1V2~]1Un1 ,n2 ,n3 ,n4

~ i j ! 2Un111,n2 ,n3 ,n4

~ i j ! !50, ~47!

]4Un1 ,n2 ,n3 ,n4

~ i j ! 2Un1 ,n2 ,n3 ,n411~ i j ! 1W1~]2

3Un1 ,n2 ,n3 ,n4

~ i j ! 23]22Un1 ,n211,n3 ,n4

~ i j ! 13]2Un1 ,n212,n3 ,n4

~ i j !

2Un1 ,n213,n3 ,n4

~ i j ! !1W2~]22Un1 ,n2 ,n3 ,n4

~ i j ! 22]2Un1 ,n211,n3 ,n4

~ i j ! 1Un1 ,n212,n3 ,n4

~ i j ! !

1W3~]2Un1 ,n2 ,n3 ,n4

~ i j ! 2Un1 ,n211,n3 ,n4

~ i j ! !50, ~48!

~ i j !5~11!, ~21!, ~31!, ~32!, ~4p!, p51,2,3.

This differential-difference system of equations is a general system generated by the algebraicsystem~1! with unit right-hand side andx-parameters introduced by the formulas~37!.

Now let us impose reductions of the form~18! on the functionsf nk( j ) to decrease the number of

discrete parameters and reduce the system~47! and ~48! to the complete system of pure PDEs.Below we consider the set of such reductions which eventually leads to mKP~64!.

Reduction 1. Introduce the following relations among the functionsf( i j ):

]2f~ i j !5]1f~ i j !, ]3f~ i j !52]12f~ i j !, ]4f~ i j !5]1

3f~ i j !. ~49!

Appropriate relations among the functionsUn1 ,n2 ,n3 ,n4

( i j ) read

Un1 ,n2 ,n3 ,n4

~ i j ! 5~21!n3Un11n212n313n4,0,0,0~ i j ! [~21!n3Un11n212n313n4

~ i j ! . ~50!

Under this reduction the system~47! and ~48! becomes of the form

]3Un~ i j !1Un12

~ i j ! 1V1~]12Un

~ i j !22]1Un11~ i j ! 1Un12

~ i j ! !1V2~]1Un~ i j !2Un11

~ i j ! !50, ~51!

]4Un~ i j !2Un13

~ i j ! 1W1~]23Un

~ i j !23]22Un11

~ i j ! 13]2Un12~ i j ! 2Un13

~ i j ! !1W2~]22Un

~ i j !22]2Un11~ i j ! 1Un12

~ i j ! !

1W3~]2Un~ i j !2Un11

~ i j ! !50, ~52!

~ i j !5~11!, ~21!, ~31!, ~32!, ~4p!, p51,2,3,

where

V152U0~32!/U0

~11! , V25„2U0~31!2V1~2]1U0

~11!2U1~11!!…/U0

~11! ,

W152U0~43!/U0

~21! , W25„2U0~42!2W1~3]2U0

~21!2U1~21!!…/U0

~21! ,

W35„2U0~41!2W1~3]2

2U0~21!23]2U1

~21!1U2~21!!2W2~2]2U0

~21!2U1~21!!…/U0

~21! .

The complete system of nonlinear PDEs is represented by the system~51! and~52! where indexnruns the values 0, 1, 2 in Eq.~51! and 0, 1 in Eq.~52!.

Reduction 2. Impose another reduction:

f~11![f~1!, f~21![f~2!, f~31!52]1f~32![2]1f~3!,~53!

f~41!5]12f~43![]1

2f~4!, f~42!52]1f~43![2]1f~4!,



129.105.215.146 On: Sun, 21 Dec 2014 15:38:36

or

Un~11![Un

~1! , Un~21![Un

~2! , Un~31!52Un11

~32! [2Un11~3! ,

~54!Un

~41!5Un12~43! [Un12

4 , Un~42!52Un11

~43! [2Un11~4! ,

which reduces the system~51! and ~52! into the following one:

]3Un~ i !1Un12

~ j ! 1V1~]12Un

~ i !22]1Un11~ i ! 1Un12

~ i ! !1V2~]1Un~ i !2Un11

~ i ! !50, ~55!

]4Un~ i !2Un13

~ i ! 1W1~]23Un

~ i !23]22Un11

~ i ! 13]2Un12~ i ! 2Un13

~ i ! !1W2~]22Un

~ i !22]2Un11~ i ! 1Un12

~ i ! !

1W3~]2Un~ i !2Un11

~ i ! !50, i 51,2,3,4, ~56!

where

V152U0~3!/U0

~1! , V25„U1~3!2V1~2]1U0

~1!2U1~1!!…/U0

~1! ,

W152U0~4!/U0

~2! , W25„U1~4!2W1~3]2U0

~2!2U1~2!!…/U0

~2! ,

W35„2U2~4!2W1~3]2

2U0~2!23]2U1

~2!1U2~2!!2W2~2]2U0

~2!2U1~2!!…/U0

~2! .

The system~55! and ~56! represents the complete system of nonlinear PDEs if indexn runs thevalues 0, 1, 2 in Eq.~55! and 0, 1 in Eq.~56!.

Reduction 3. Assume now

f~3!5f~1!, f~4!5f~2!, or Un~3!5Un

~1! , Un~4!5Un

~2! . ~57!

The system~55! and ~56! becomes of the form

]3Un~ i !2]1

2Un~ i !12]1Un11

~ i ! 1V2~]1Un~ i !2Un11

~ i ! !50, ~58!

]4Un~ i !2]2

3Un~ i !13]2

2Un11~ i ! 23]2Un12

~ i ! 1W2~]22Un

~ i !22]2Un11~ i ! 1Un12

~ i ! !1W3~]2Un~ i !2Un11

~ i ! !

50, i 51,2, ~59!

where

V252]1U0~1!/U0

~1! , W253]1U0~2!/U0

~2! ,

W35„3]22U0

~2!23]2U1~2!2W2~2]2U0

~2!2U1~2!!…/U0

~2! .

This system represents the complete system of PDEs ifn50,1 in Eq.~58! andn50 in Eq. ~59!.Reduction 4. The next possible reduction is

f~1!5f~2!5f or Un~1!5Un

~2!5Un. ~60!

Then the system~58! and ~59! becomes of the form

]3Un2]12Un12]1Un111V2~]1Un2Un11!50, ~61!

]4Un2]23Un13]2

2Un1123]2Un121W2~]22Un22]2Un111Un12!1W3~]2Un2Un11!50,

~62!

where



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V252]1U0 /U0 , W253]2U0 /U0 , W35„3]22U023]2U12W2~2]2U02U1!…/U0 .

This system13 represents the complete system of PDEs ifn50,1 in Eq.~61! andn50 in Eq.~62!.It can be rewritten as

ux32ux1x1

12vx11ux1

2 50, vx32vx1x1

12wx122vvx1

50,

~63!ux4

2ux2x2x213vx2x2

23wx213ux2

ux2x22ux2

3 23ux2vx2

13vvx250,

whereU05expu, U15vU0 , andU25wU0 .Reduction 5. Finally,13 the reduction]2[]1 reduces the (311)-dimensional system~63! into

mKP:

]4v21/4 ]13v13/2 v2]1v13/2 ]1v]1

21]3v23/4 ]121]3

2v50, v5]1U0 /U0 . ~64!

In this way we have demonstrated that the general system~47! and ~48! can be regarded as thedifferential-difference generalization of mKP with four continuous and four discrete parameters.

A. Construction of particular solutions

As we mentioned earlier, the main problem in construction of the particular solutions is thecompatibility conditions~6! or ~19!. We start with the solutions to the general differential-difference system~47! and ~48!. The condition~19! is represented by the system of three equa-tions:

]2f m~1!gm

~1!1 f n1~1!]2gm

~1!2]1f n1~2!gm

~2!2 f n1~2!]1gm

~2!50, ~65!

~]3f n1~1!2]1f n1

~3!!gm~1!2~ f n1

~3!1]1f n2~3!!]1gm

~1!1 f n1~1!]3gm

~1!2 f n2~3!]1

2gm~1!50, ~66!

~]4f n1~2!2]2f n1

~4!!gm~2!2~ f n1

~4!1]2f n2~4!!]2gm

~2!2~ f n2~4!1]2f n3

~4!!]22gm

~2!1 f n1~2!]4gm

~2!2 f n3~4!]2

3gm~2!50,

~67!

so that one hasx15x25(x1 ,x2 ,x3 ,x4). This system can be considered as the linear system on thefunctions f nk

( j ) . It is not difficult to understand that this system is resolvable forN(r )5N(c)51 ~orn5m51, step1!, when it can be treated as the system on the functionsf 11

( j ) , j 52,3,4. All otherfunctions can be taken as arbitrary ones. It means that the whole nonlinear system~47! and ~48!can be treated by the dressing method.

Let us fix Nmax(c) 51 and show that the appropriateNmax

(r) 5`. In fact, for each particularn onecan solve the system~65!–~67! for the functionsf n1

( j ) , j 52,3,4. It means thatf- andg-functionsbelong to the manifoldsF1`

( i ) andG11( i ) with arbitrary functionsg1

( i ) ( i 51,2), f n1(1) , f n2

( j ) ( j 53,4), f n3( j )

( j 54), n51,2,....Similarly, the system~65!–~67! is consistent forN(c)52 ~step 2!. In fact, in this casegm

( j )

(m51,2) can be taken as arbitrary functions ofx. Then for each particular value of the parametern the whole system is the linear system of six PDEs for seven functionsf nk

( j ) , ( jk)5(1,1),(2,1),(3,1),(3,2),(4,1),(4,2),(4,3). One of these functions~say f n1

(4)! can be arbitrary. So,each of the equations~65!–~67! is a PDE whose solution depends on the arbitrary functionsgm

( j )

(m51,2) andf n1(4) . This means thatNmax

(r) 5`. From this we conclude thatg-functions form mani-folds G22

( i ) , while f-functions form manifoldsF1`( i ) .

The analysis of the caseN(c)53 leads to thenonlinear four-dimensional PDE onf- andg-functions.

Let us consider the set of particular solutions in more detail, related with one of the repre-sentations~26! or ~27! of the f- andg-functions~step 3!. We take the representation~26!, whichreads in our case



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f nm~ i ! 5anm

~ i ! e2knx12knx22kn2x32kn

3x4, ~68!

gm~ i !5E bm

~ i !~vm1 ,vm2 ,vm3 ,vm4!evm1x11vm2x21vm3x31vm4x4 dvm1dvm2dvm3dvm4 . ~69!

This representation reduces the system~65!–~67! into the system of algebraic equations. Belowwe discuss particular solutions to the nonlinear PDEs~47! and ~48! with reductions 1–4.

Consider the system~51! and ~52! associated withreduction 1. The equations~65!–~67!become of the form

an1~1!bm

~1!~kn2vm2!1an1~2!bm

~2!~vm12kn!50, ~70!

an1~1!~kn

22vm3!1~an1~3!1an2

~3!vm1!~vm12kn!50, ~71!

an1~2!~kn

32vm4!1~an1~4!1an2

~4!vm21an3~4!vm2

2 !~vm22kn!50. ~72!

The following families of particular solutions are available:~1! ~Criteria!. If N(c)5N(r )51 in ~70!–~72!, the system is resolvable forb1

(2) , v13, andv14,so thatf- andg-functions belong to the manifoldsF11

( i ) andG11( i ) , respectively, and~69! becomes of

the form

g1~ i !5E b1

~ i !~v11,v12!ev11x11v12x21v13x31v14x4 dv11dv12, ~73!

wherev13 andv14 are functions ofv11 andv12.~2! Let us take the system~70!–~72! with n51 (Nmin

(r) 51) to fix relations among the param-etersbm

( j ) ( j 51,2), vmk (k51,2,3,4):

bm~2!5

a11~1!bm

~1!~k12vm2!

a11~2!~k12vm1!

, ~74!

vm35~k12a11

~1!2k1a11~3!1~a11

~3!2k1a12~3!!vm11a12

~3!vm12 !/a11

~1! , ~75!

vm45~k13a11

~2!2k1a11~4!1~a11

~4!2k1a12~4!!vm21~a12

~4!2k1a13~4!!vm2

2 1a13~4!vm2

3 !/a11~2! . ~76!

Here vm1 , vm2 are integration parameters in Fourier representation~73!, which means thatNmin

(c) 51 and Nmax(c) 5`. One can check that the system~70!–~72! does not have solutions with

differentknÞk1 , i.e.,Nmax(r) 51. So,f- andg-functions are collected in the manifoldsF11

( i ) andG1`( i ) ,

respectively.~3! Consider~70!–~72! with n51,2 (Nmin

(r) 52) to expressbm(2) , vm j , j 51,2,3, throughvm1

andbm(1) . Then along with~74!–~76! one has from the equation~70!

vm25k1k2~a21

~1!a11~2!2a11

~1!a21~2!!1~q1a11

~1!a21~2!2q2a21

~1!a11~2!!vm1

~q12vm1!a21~1!a11

~2!2~q22vm1!a11~1!a21

~2! , ~77!

from the equation~71!

a12~3!5„~k11k2!a11

~1!2a11~3!…/k2 , ~78!

a22~3!5

a21~1!

a11~1! a12

~3! , ~79!



129.105.215.146 On: Sun, 21 Dec 2014 15:38:36

a21~3!5

a21~1!

k2a11~1! „~k2

22k12!a11

~1!1k1a11~3!…, ~80!

and from the equation~72!

a13~4!5

1

k22 ~a11

~2!~k121k1k21k2

2!2a11~4!2k2a12

~4!!, ~81!

a23~4!5a21

~2!a13~4!/a11

~2! , ~82!

a22~4!5~a21

~2!a13~4!~k22k1!1a21

~2!a12~4!!/a11

~2! , ~83!

a21~4!5a21

~2!~a11~4!1~k22k1!a12

~4!1a13~4!k2~k22k1!!/a11

~2!. ~84!

One can check that the system~70!–~72! together with formulas~74!–~84! has no solutions withknÞk1 ,k2 . This means thatNmin

(r) 5Nmax(r) 52, Nmin

(c) 51, Nmax(c) 5`, f- and g-functions belong to the

manifolds F22( i ) and G1`

( i ) , and vm1 is an integration parameter in Fourier representation ofg-functions:

gm~ i !5E bm

~ i !~vm1!evm1x11vm2x21vm3x31vm4x4 dvm1 . ~85!

~4! Consider Eqs.~70!–~72! with n51,2,3 to fix the parametersbm(2) , vm j , j 51,2,3,4. One

gets

vm15„k1k2a31~1!~a21

~1!a11~2!2a11

~1!a21~2!!1k1k3a21

~1!~a11~1!a31

~2!2a31~1!a11

~2!!1k2k3a11~1!~a31

~1!a21~2!

2a21~1!a31

~2!!…„a21~1!a31

~1!a11~2!~q22q3!1a11

~1!a31~1!a21

~2!~q32q1!1a11~1!a21

~1!a31~2!~q12q2!…21

~86!

along with~74!–~77!. Then the equations~70!–~72! with n.3 admit an infinite number of solu-tions, parametrized, for example, byan1

(1) , kn . The dimensionsNmin(c) 5Nmax

(c) 51, Nmin(r) 53, Nmax

(r)

5` and integration in~85! disappears:

gm~ i !5bm

~ i !evm1x11vm2x21vm3x31vm4x4. ~87!

Heref- andg-functions belong to the manifoldsF3`( i ) andG11

( i ) . In this case the representation~68!for f-functions can be replaced by the Fourier integral@see~27!#.

Analogous families of particular solutions are available for the nonlinear systems~55! and~56!, and ~58! and ~59! related toreductions 2and 3, respectively~which mean appropriatereductions on the parametersank

( i )!.Under reduction 2one has

an1~1!5an

~1! , an1~2!5an

~2! , an1~3!5an

~3!kn , an2~3!5an

~3! ,~88!

an1~4!5an

~4!kn2, an2

~4!5an~4!kn , an3

~4!5an~4! .

The system~70!–~72! becomes of the form

an~1!bm

~1!~kn2vm2!1an~2!bm

~2!~vm12kn!50, ~89!

an~1!~kn

22vm3!1an~3!~vm1

2 2kn2!50, ~90!

an~2!~kn

32vm4!1an~4!~vm2

3 2kn3!50. ~91!



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Reduction 3gives

an~3!5an

~1! , an~4!5an

~2! ~92!

and one has

an~1!bm

~1!~kn2vm2!1an~2!bm

~2!~vm12kn!50, ~93!

vm12 2vm350, vm2

3 2vm450 ~94!

instead of Eqs.~89!–~91!.Reduction 4(an

(1)5an(2)) keeps Eq.~94! unchanged and reduces Eq.~93! into the following

one:

bm~1!~kn2vm2!1bm

~2!~vm12kn!50, ~95!

which has been considered in Ref. 13. It admits the family of solutions withNmin(c) 5Nmax

(c) 51,Nmin

(r) 51, Nmax(r) 5`, where vm1 , vm2 are integration variables in~69!. f- and g-functions are

collected in the manifoldsF11( i ) andG1`

( i ) .Finally, reduction 5assumesbm

(1)5bm(2) , vm15vm2 and reduces the system of nonlinear PDEs

~63! into the completely integrable equation mKP~64!.

IV. RELATION BETWEEN DIFFERENT INTEGRABLE HIERARCHIES

The dressing method based on the algebraic system~1! allows one to establish the relationsbetween solutions of different integrable hierarchies. To demonstrate this let us consider two setsof parametersx5(x1 ,...) andy5(y1 ,...) which are independent variables in two different inte-grable hierarchies of nonlinear PDEs. These parameters are introduced by the system which issimilar to the system~5!:

]xiR5 (

k51

Nxi

f~xi ,k!g~k,xi !, ]yiR5 (

k51

Nyi

f~yi ,k!g~k,yi !, ~96!

fzi ,k5@ f 1k~zi !

¯ fN~r !k

~zi ! #T, gk,zi5@gk1~zi !

¯gkN~c!

~zi ! #, ~97!

where zi is either xi or yi . Since each of the equations~96! serves an integrable system, thecompatibility conditions (]xixj

2]xjxi)R50 and (]yiyj

2]yj yi)R50 are satisfied so that the only

compatibility condition which should be considered is the following one@see note above the Eq.~19!#:

~]x1y12]y1x1

!R50. ~98!

The equation~98! gives rise to the relations amongf- andg-functions.Let

Mn~1!c50, Mm

~2!c50, n51,..., m51,..., c5@c1 ...cN~c!# ~99!

be the linear overdetermined systems related with two hierarchies pointed out earlier. The set offunctions U(1)5$Un1...

(1ik)5cC()s]xs

ns)f(xi ,k)% satisfies the first hierarchy of nonlinear equations,

while the set of functionsU(2)5$Un1...

(2ik)5cC()s]ys

ns)f(yi ,k)% satisfies the second hierarchy of non-

linear equations. The relation between these two hierarchies is described by the nonlinear systemon the ‘‘mixed’’ functionsU5$Un1...

(12ik) ,Un1...

(21ik)%, where



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Un1 ,...,m1 ,...~12ik ! 5cCS)

s]xs

nsD S)r

]yr

mr D f~xi ,k!, Un1 ,...,m1 ,...~21ik ! 5cCS)

s]xs

nsD S)r

]yr

mr D f~yi ,k!.

As an example let us consider two different mKP hierarchies, associated with two differentsets of independent variables:x5(x1 ,...) andy5(y1 ,...). Thedependence on these variables isintroduced by the following equations:

]x1R5f~1!g~1!, ]x2

R5f~1!]x1g~1!2]x1

f~1!g~1!, ~100!

]x3R5f~1!]x1

2 g~1!2]x1f~1!]x1

g~1!1]x1

2 f~1!g~1!, ~101!

]y1R5f~2!g~2!, ]y2

R5f~2!]y1g~2!2]y1

f~2!g~2!, ~102!

]y3R5f~2!]y1

2 g~2!2]y1f~2!]y1

g~2!1]y1

2 f~2!g~2!, ~103!

]xjg~1!5]x1

j g~1!, ]xjf~1!5~21! j]x1

j f~1!, ~104!

]yjg~2!5]y1

j g~2!, ]yjf~2!5~21! j]y1

j f~2!, ~105!

f~ i !5@ f 1~ i !¯ f N~r !

~ i !#T, g~ i !5@g1

~ i !¯gN~c!

~ i !#, i 51,2. ~106!

The compatibility condition~98! has the form

]y1~ f~1!g~1!!5]x1

~ f~2!g~2!!. ~107!

The investigation of this condition is analogous to the investigation of the conditions~65!–~67!.We do not represent it here. The linear system associated with parametersx1 ,x2 ,x3 andy1 ,y2 ,y3

follows:

]x2c2]x1

2 c1V1~1!]x1

c50, ~108!

]x3c2]x1

3 c1W1~1!]x1

2 c1W2~1!]x1

c50, ~109!

]y2c2]y1

2 c1V1~2!]y1

c50, ~110!

]y3c2]y1

3 c1W1~2!]y1

2 c1W2~2!]y1

c50, ~111!

where

V1~1!52]x1

U0,0~1!/U0,0

~1! , W1~1!53]x1

U0,0~1!/U0,0

~1! , ~112!

W2~1!5„3]x1

2 U0,0~1!23]x1

U1,0~1!2W1

~1!„2]x1

U0,0,~1!2U1,0

~1!)…/U0,0~1! ~113!

V1~2!52]y1

U0,0~2!/U0,0, W1

~2!53]y1U0,0

~2!/U0,0~2! , ~114!

W2~2!5„3]y1

2 U0,0~2!23]y1

U0,1~2!2W1

~2!~2]y1U0,0

~2!2U0,1~2!!…/U0,0

~2! , ~115!

Un,m~ j ! 5cC]x1

n ]y1

m f~ j !, j 51,2.



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The system of nonlinear PDEs, which relates two hierarchies, results from the equations~108! and~110! if one multiplies them by the vectorsC]x1

n ]y1

m f( i ) (f( i )5@ f 1( i ) , ...,f N(r )

( i )#T, i 51,2):

]x2Un,m

~ i ! 2]x1

2 Un,m~ i ! 12]x1

Un11,m~ i ! 1V1

~1!~]x1Un,m

~ i ! 2Un11,m~ i ! !50, n50, m50,1, ~116!

]y2Un,m

~ i ! 2]y1

2 Un,m~ i ! 12]y1

Un,m11~ i ! 1V1

~2!~]y1Un,m

~ i ! 2Un11,m~ i ! !50, n50,1, m50. ~117!

Possible reductions are the following ones,

f~1!5f~2!, or Un,m~1! 5Un,m

~2! , ~118!

and reductions which establish the relation between the discrete parametersn andm. For example,the reduction

]y1f~ i !5]x1

f~ i ! ~119!

eliminates one of the discrete parameters in the system~116! and ~117!.

V. POSSIBLE GENERALIZATIONS OF THE DRESSING METHOD

We give some remarks about two generalizations of the algorithm represented in this paper.

~1! Similarly to the ]-problem~2!, which admits different normalization functionsh,6 the alge-braic system~1! also admits different functionsh on the right-hand side. For example, thealgebraic system~1! can be replaced with the following one:L(c (nm))5g(nm). Generalizationof KP is an example of the related system of nonlinear PDEs~see the Appendix!.

~2! ~Discrete version of the matrix]-problem.7! Instead of the scalar system ofN(c) equations~1!,one can take matrixK13K2 algebraic system ofN(c) equations~or tensor equation!. In thiscase each entry of the matricesC andR is represented byK13K ~K is an arbitrary integer!andK3K2 matrices, respectively, and the system~5! which introduces the independent vari-ablesx reads:] iRnm5Sk51

Ni fnk( i )gkm

( i ) , ~n51,...,N(r ), m51,...,N(c)!, where fnk( i ) are K3K1 and

gkm( i ) areK13K2 matrices for all possible values of the indexesi,k,n,m. All scalar equations of

the previous sections become theK13K2 matrix equations. This generalization is beyond thescope of this article.

VI. CONCLUSIONS

The introduced modification of the dressing method is aimed at the construction of newclasses of either integrable or nonintegrable equations together with families of particular solu-tions. The families of particular solutions are parametrized either by the arbitrary functions ofindependent variables or constant parameters. The maximum possible number of these arbitraryfunctions~or parameters! can be taken as the characteristic of the related nonlinear system. An-other characteristic is the maximum possible dimensionsN(r ) and/orN(c) of the matrixR in thealgebraic system~1!. The introduced nonintegrable generalizations of the integrable systems ofnonlinear PDEs can be applied in studies of the physical phenomena in systems with smallparameters.

Another question which should be studied is the relation of this method with the Hirotamethod.15–18 In both cases one relates the original nonlinear PDEs with the bilinear system ofequations@Eqs. ~6! in our case#. But in our casef- and g-functions are also subjected to theequations~9! and ~18!, which put additional restrictions on these functions.

ACKNOWLEDGMENTS

The author thanks Academician V. E. Zakharov and Professor S. V. Manakov for helpfuldiscussions, and the referee for useful comments.



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APPENDIX: GENERALIZATION OF KP

In this section we consider the example of the (211)-dimensional system of nonlinear PDEs,related with the first generalization of the dressing method, mentioned in Sec. V. Namely, let usconsider the linear algebraic system of the form

L~c~p!!5g~p!, p51,2, c~p!5@c1~p!

¯cN~c!~p!

#, g~p!5@g1~p!

¯gN~c!~p!

#, ~A1!

with M53. Introduce variablesx1 ,x2 ,x3 by the following formulas:

]1R5f~11!g~1!, ]2R5f~21!g~2!1f~22!]1g~2!,~A2!

]3R5f~31!g~3!1f~32!]1g~3!1f~33!]12g~3!, f~nk!5@ f 1k

~n!¯gN~r !k

~n!#T.

The algebraic system~A1! and equations~A2! generate the following set of algebraic systems withdifferent nonhomogeneous parts@compare with Eqs.~11!–~13!#:

L~]1c~p!!5U0,0,0~p11!g~1!1]1g~p!, ~A3!

L~]12c~p!!5~2]1U0,0,0

~p11!2U1,0,0~p11!!g~1!1U0,0,0

~p11!]1g~1!1]12g~p!, ~A4!

L~]13c~p!!5~3]1

2U0,0,0~p11!23]1U1,0,0

~p11!1U2,0,0~p11!!g~1!1~3]1U0,0,0

~p11!2U1,0,0~p11!!]1g~1!1U0,0,0

~p11!]12g~1!

1]13g~p!, ~A5!

L~]2c~p!!5U0,0,0~p21!g~2!1U0,0,0

~p22!]1g~2!1]12g~p!, ~A6!

L~]3c~p!!5U0,0,0~p31!g~3!1U0,0,0

~p32!]1g~3!1U0,0,0~p33!]1

2g~3!1]13g~p!, ~A7!

Un1 ,n2 ,n3

pnk 5c~p!C]1n1]2

n2]3n3f~nk!. ~A8!

By using the set of systems~A1! and~A3!–~A7!, one can construct the linear system~15! of PDEson the functionsc (p) which has the form

]2c~p!2]12c~p!1V1

~p!]1c~1!1V2~p!]1c~2!1V3

~p!c~1!1V4~p!c~2!50, ~A9!

]3c~p!2]13c~p!1W1

~p!]12c~1!1W2

~p!]12c~3!1W3

~p!]1c~1!1W4~p!]1c~3!1W5

~p!c~1!1W6~p!c~3!50,

~A10!

whereVi andWi are related with functionsUn1 ,n2 ,n3

(pi j ) by the following formulas:

V1~p!2U0,0,0

~p11!50, V2~p!1U0,0,0

~p22!50,

V3~p!2~2]1U0,0,0

~p11!2U1,0,0~p11!!1V1

~p!U0,0,0~111!1V2

~p!U0,0,0~211!50,

V4~p!1U0,0,0

~p21!50, W1~p!2U0,0,0

~p11!50, W2~p!1U0,0,0

~p33!50,

W3~p!2~3]1U0,0,0

~p11!2U1,0,0~p11!!1W1

~p!U0,0,0~111!1W2

~p!U0,0,0~311!50, W4

~p!1U0,0,0~p32!50, ~A11!

W5~p!2~3]1

2U0,0,0~p11!23]1U0,0,0

~p11!1U2,0,0~p11!!1W1

~p!~2]1U0,0,0~111!2U1,0,0

~111!!1W2~p!~2]1U0,0,0

~311!2U1,0,0~311!!

1W3~p!U0,0,0

~111!1W4~p!U0,0,0

~311!50, W6~p!1U0,0,0

~p31!50.



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The appropriate nonlinear system on the functionsUn1 ,n2 ,n3

pnk ~A8! can be received by multiplying

each equation of the above linear system~A9! and~A10! ~which are 13N(c) matrix equations! bythe N(c)31 vectorC]1

n1]2n2]3

n3f(nk). One obtains

]2Un1 ,n2 ,n3

~pkn! 2Un1 ,n211,n3

~pkn! 2]12Un1 ,n2 ,n3

~pkn! 12]1Un111,n2 ,n3

~pkn! 2Un112,n2 ,n3

~pkn! 1V1~p!~]1Un1 ,n2 ,n3

~1kn!

2Un111,n2 ,n3

~1kn! !1V2~p!~]1Un1 ,n2 ,n3

~2kn! 2Un111,n2 ,n3

~2kn! !1V3~p!Un1 ,n2 ,n3

~1kn! 1V4~p!Un1 ,n2 ,n3

~2kn! 50,

~A12!

]3Un1 ,n2 ,n3

~pkn! 2Un1 ,n2 ,n311~pkn! 2]1

3Un1 ,n2 ,n3

~pkn! 13]12Un111,n2 ,n3

~pkn! 23]1Un112,n2 ,n3

~pkn! 1Un113,n2 ,n3

~pkn! 1W1~p!

3~]12Un1 ,n2 ,n3

~1kn! 22]1Un111,n2 ,n3

~1kn! 1Un112,n2 ,n3

~1kn! !1W2~p!~]1

2Un1 ,n2 ,n3

~3kn! 22]1Un111,n2 ,n3

~3kn!

1Un112,n2 ,n3

~3kn! !1W3~p!~]1Un1 ,n2 ,n3

~1kn! 2Un111,n2 ,n3

~1kn! !1W4~p!~]1Un1 ,n2 ,n3

~3kn! 2Un111,n2 ,n3

~3kn! !

1W5~p!Un1 ,n2 ,n3

~1kn! 1W6~p!Un1 ,n2 ,n3

~3kn! 50, ~A13!

~pkn!5~p11!, ~p31!, ~p32!, ~p4s!, p51,2, s51,2,3.

This differential-difference system involves three continuous and three discrete parameters. Toreduce it to the system of pure nonlinear PDEs one needs to put some additional relations amongfunctions f nk

( j ) . For example, consider the followingReduction 1:

]2f~ jk !52]12f~ jk !, ]3f~ jk !5]1

3f~ jk !, ~A14!

f~21!52]1f~22!, f~32!52]1f~33!, f~31!5]12f~33!, ~A15!

or

Un1 ,n2 ,n3

~pkn! 5~21!n2Un112n213n3,0,0~pkn! [~21!n2Un112n213n3

~pkn! , ~A16!

Uk~p11![Uk

~p1! , Uk~p21!52Uk11

~p22![2Uk11~p2! , ~A17!

Uk~p32!52Uk11

~p33![2Uk11~p3! , Uk

~p31!5Uk12~p33![Uk12

~p3!. ~A18!

Then the system@~A12!,~A13!# becomes of the form

]2Ukp j2]1

2Ukp j12]1Uk11

p j 1U0~p1!~]1Uk

~1 j !2Uk11~1 j ! !2U0

~p2!~]1Uk~2 j !2Uk11

~2 j ! ! ~A19!

1~2]1U0~p1!2U1

~p1!2U0~p1!U0

~11!1U0~p2!U0

~21!!Uk~1 j !1U1

~p2!Uk~2 j !50, k50,1,

]3Uk~p j !2~]1

3Uk~p j !23]1

2Uk11~p j ! 13]1Uk12

~p j ! !1W1~p!~]1

2Uk~1 j !

22]1Uk11~1 j ! 1Uk12

~1 j ! !1W2~p!~]1

2Uk~3 j !22]1Uk11

~3 j ! 1Uk12~3 j ! ! ~A20!

1W3~p!~]1Uk

~1 j !2Uk11~1 j ! !1W4

~p!~]1Uk~3 j !2Uk11

~3 j ! !1W5~p!Uk

~1 j !1W6~p!Uk

~3 j !50,

p51,2, j 51,2,3,

W1~p!5U0

~p1!, W2~p!52U0

~p3!,

W3~p!53]1U0

~p1!2U1~p1!2U0

~p1!U0~11!1U0

~p3!U0~31!, W4

~p!5U1~p3!,

W5~p!53]1

2U0~p1!23]1U1

~p1!1U2~p1!2U0

~p1!~2]1U0~11!2U1

~11!!



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1U0~p3!~2]1U0

~31!2U1~31!!2~3]1U0

~p1!2U1~p1!2U0

~p1!U0~11!

1U0~p3!U0

~31!!U0~11!2U1

~p3!U0~31!, W6

~P!52U2~p3!.

Using anotherReduction 2

f~11!5f~22!5f~33![f, g~11!5g~21!5g~31![g, or Uk~p j ![Uk ~A21!

one reduces the nonlinear system@~A19!,~A20!# into KP on the functionu5]1U0:

ux32 1

4ux1x1x113uux1

- 34]x1

21ux2x250. ~A22!

Different Reduction 3

f~11!5f~33!52f~22![f, g~11!5g~21!5g~31![g ~A23!

reduces the nonlinear system@~A19!,~A20!# into the following nonintegrable one

]2Uk2]12Uk12]1Uk1112U0~]0Uk2Uk11! ~A24!

12~]1U02U12U0U0!Uk50, k50,1 ~A25!

]3U02]13U013]1

2U123]1U213]1U0~]1U02U1!13~]12U02]1U1!U023U0U0]1U050.

One can demonstrate that the compatibility condition~6! can be satisfied~at least forN(c)

5N(r )51) for all nonlinear equations considered in the Appendix, i.e., the dressing method isapplicable to them.

1V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl.8, 43 ~1974!.2V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl.13, 13 ~1979!.3M. J. Ablowitz and H. Segur,Solitons and Inverse Scattering Transform~SIAM, Philadelphia, 1981!.4V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevsky,Theory of Solitons. The Inverse Problem Method~Plenum, New York, 1984!.

5V. E. Zakharov and S. V. Manakov, Funct. Anal. Appl.19, 11 ~1985!.6L. V. Bogdanov and S. V. Manakov, J. Phys. A21, L537 ~1988!.7B. Konopelchenko,Solitons in Multidimensions~World Scientific, Singapore, 1993!.8V. E. Zakharov and S. V. Manakov, Theor. Math. Phys.27, 283 ~1976!.9S. V. Manakov, Usp. Mat. Nauk31, 245 ~1976! ~in Russian!.

10P. A. Clarkson, P. R. Gordoa, and A. Pickering, Inverse Probl.13, 1463~1997!.11I. A. B. Strachan, Inverse Probl.8, L21 ~1992!.12A. I. Zenchuk, J. Math. Phys.41, 6248~2000!.13A. I. Zenchuk, Phys. Lett. A277, 25 ~2000!.14The representation of the]-problem by the system of equations~2!–~4! defers from the standard representation.8,7 This

representation has been found in collaboration with Professor S. V. Manakov and is more convenient for our purpose. Wedo not discuss the system~2!–~4! since it is written here only to explain the origin of the algebraic system~1! and willnot be used in this article.

15R. Hirota, in Lecture Notes in Mathematics, Vol. 515~Springer-Verlag, New York, 1976!.16R. Hirota, J. Phys. Soc. Jpn.46, 312 ~1979!.17R. Hirota and J. Satsuma, Suppl. Prog. Theor. Phys.59, 64 ~1976!.18R. Hirota and J. Satsuma, J. Phys. Soc. Jpn.40, 891 ~1976!.



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Families of particular solutions to multidimensional partial differential equations