whitham systems and deformations

Whitham systems and deformationsA. Ya. Maltsev Citation: Journal of Mathematical Physics 47, 073505 (2006); doi: 10.1063/1.2217648 View online: http://dx.doi.org/10.1063/1.2217648 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/47/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Alternative sets of multisoliton solutions of some integrable KdV type equations via direct methods AIP Conf. Proc. 1479, 1369 (2012); 10.1063/1.4756411 Stable difference scheme for nonlocal boundary value hyperbolic problems involving multi-point and integralconditions AIP Conf. Proc. 1479, 602 (2012); 10.1063/1.4756204 On stability of hyperbolic-elliptic differential equations with nonlocal integral condition AIP Conf. Proc. 1470, 73 (2012); 10.1063/1.4747642 NBVP for hyperbolic equations involving multi-point and integral conditions AIP Conf. Proc. 1470, 65 (2012); 10.1063/1.4747640 Darboux-integrable two-component nonlinear hyperbolic systems of equations J. Math. Phys. 52, 033503 (2011); 10.1063/1.3559134

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Whitham systems and deformationsA. Ya. Maltseva�

L.D.Landau Institute for Theoretical Physics, 119334 ul. Kosygina 2, Moscow, Russia

�Received 20 September 2005; accepted 1 June 2006; published online 21 July 2006�

We consider the deformations of Whitham systems including the “dispersionterms” and having the form of Dubrovin-Zhang deformations of Frobenius mani-folds. The procedure is connected with the B. A. Dubrovin problem of deforma-tions of Frobenius manifolds corresponding to the Whitham systems of integrablehierarchies. Under some nondegeneracy requirements we suggest a general schemeof the deformation of the hyperbolic Whitham systems using the initial nonlinearsystem. The general form of the deformed Whitham system coincides with the formof the “low-dispersion” asymptotic expansions used by B. A. Dubrovin andY. Zhang in the theory of deformations of Frobenius manifolds. © 2006 AmericanInstitute of Physics. �DOI: 10.1063/1.2217648�

I. INTRODUCTION

The classical Whitham method1–4 is connected with the slow modulations of the exact peri-odic or quasiperiodic solutions of nonlinear PDE’s,

Fi��,�t,�x,�tt,�xt,�xx, . . . � = 0, i = 1, . . . ,n , �1.1�

where �= ��1 , . . . ,�n�.It is assumed that the system �1.1� admits the finite-parametric family of exact solutions

�i�x,t� = �i�k�U�x + ��U�t + �0,U� , �1.2�

where �= ��1 , . . . ,�m�, �i�� ,U� are smooth functions 2�-periodic with respect to each ��, k�U�= �k1�U� , . . . ,km�U��, ��U�= ��1�U� , . . . ,�m�U�� are “wave numbers” and “frequencies” of thesolution, U= �U1 , . . . ,UN� are parameters of the solution, and �0= ��0

1 , . . . ,�0m� are arbitrary initial

phases.The functions �� ,U� satisfy the nonlinear system

Fi��,��U��,k��U��,��U��U��, . . . � = 0. �1.3�

We can introduce the families k,� and the full family = �k,� of the functions �� ,U�satisfying the system �1.3� in the space of 2�-periodic with respect to each �� function. Let uschoose �in a smooth way� at every �U1 , . . . ,UN� some function �� ,U� as having “zero initialphase shifts” and represent the full family of m-phase solutions of system �1.1� in the form �1.2�.

In Whitham method we make a rescaling X=�x, T=�t ��→0� of both variables x and t and tryto find a function

S�X,T� = �S1�X,T�, . . . ,Sm�X,T�� 1.4�

and 2�-periodic functions

a�Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS 47, 073505 �2006�

47, 073505-10022-2488/2006/47�7�/073505/18/$23.00 © 2006 American Institute of Physics


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�i��,X,T,�� = �k 0

��k�i ��,X,T��k �1.5�

such that the functions

�i��,X,T,�� = �i�S�X,T��

+ �,X,T,�� 1.6�

satisfy the system

Fi��,��T,��X,�2�TT, . . . � = 0 �1.7�

at every X, T, and �.It is easy to see that the function ��0�� ,X ,T� satisfies the system �1.3� at every X and T with

k� = SX�, �� = ST

�

and so belongs at every �X ,T� to the family . We can write then

��0�i ��,X,T� = �i�� + �0�X,T�,U�X,T��

and introduce the functions U��X ,T�, �0��X ,T� as the parameters characterizing the main term in

�1.5� which should satisfy the condition

�k��U��T = ��U��X. �1.8�

The functions ��1�i �� ,X ,T� are defined from the linear system

Lji��1�

j ��,X,T� = f �1�i ��,X,T� �1.9�

where

Lji = L�X,T�j

i =�Fi

�� j ��0��,X,T�, . . . � +�Fi

��tj ��0��,X,T�, . . . ��X,T�

�

��

+�Fi

��xj ��0��,X,T�, . . . �k��X,T�

�

�� + ¯ �1.10�

is the linearization of system �1.3� and f�1�� ,X ,T� is discrepancy given by

f �1�i ��,X,T� = −

�Fi

��tj ��0��,X,T�, . . . ��0�T

j ��,X,T� −�Fi

��xj ��0��,X,T�, . . . ��0�X

j ��,X,T�

−�Fi

��ttj ��0��,X,T�, . . . ��2��X,T��0��T

j + �T��X,T��0��

j � − . . . . �1.11�

We have here

�

�T= UT

� �

�U� + ��0�T� �

�� ,�

�X= UX

� �

�U� + ��0�X� �

��

for the functions

��0�i ��,X,T� = �i�� + �0�X,T�,U�X,T�� .

We will assume that k� and �� can be considered �locally� as the independent parameters onthe family and the total family of solutions of �1.3� depends �for generic k�, �� on N=2m+s, �s 0� parameters U� and m initial phases �0

�.

073505-2 A. Ya. Maltsev J. Math. Phys. 47, 073505 �2006�


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It is easy to see that the functions ��+�0�X ,T� ,U�X ,T�� and��+�0�X ,T� ,U�X ,T�� where � is any vector in the space of parameters U� tangential to the

surface k=const, �=const belong to the kernel of the operator L�X,T�ji .

Let us put now some “regularity” conditions on the family �1.2� of quasiperiodic solutions of�1.1�

Definition 1.1: We call the family (1.2) the full regular family of m-phase solutions of (1.1) if:(1) The functions �� ,U�, �U�� ,U� are linearly independent (almost everywhere) on the

set ;(2) The m+s linearly independent functions �� ,U�, �� ,U� ��k=0,��=0� give the

full kernel of the operator L�U�ji (here �0=0) for generic k and �.

(3) There are exactly m+s linearly independent “left eigen-vectors” ��U��q� ��, q=1, . . . ,m+s of

the operator L�U�ji (for generic k and �) corresponding to zero eigenvalues, i.e.,

�0

2�

. . . �0

2�

��U�i�q� ��L�U�j

i � j��dm�

�2��m 0

for any periodic � j��.It is not difficult to see that the Definition 1.1 is connected with the regularity properties of the

submanifold given by the set of functions �� ,U� in the space of 2�-periodic functions. In fact,for our purposes we can use also the weaker definition of the full regular family of m-phasesolutions of �1.1�. Namely, let us represent the space of parameters U in the form U= �k ,� ,n�where k are the wave numbers, � are the frequencies of m-phase solutions, and n= �n1 , . . . ,ns� aresome additional parameters �if they exist�. Let us give now the “weak” definition of the fullregular family of m-phase solutions in the form:

Definition 1.1�: We call the family the full regular family of m-phase solutions of (1.1) if(1) The functions �� ,k ,� ,n�, �nl�� ,k ,� ,n� are linearly independent and give (for ge-

neric k and �) the full basis in the kernel of the operator Lj��0,k,�,n�i ;

(2) The operator Lj��0,k,�,n�i has (for generic k and �) exactly m+s linearly independent “left

eigenvectors”

��U��q� �� + �0� = ��k,�,n�

�q� �� + �0�

depending on the parameters U in a smooth way and corresponding to zero eigenvalues.We will assume now that the system �1.1� has a full regular family of quasiperiodic m-phase

solutions in strong or weak sense in all our considerations.To find the function ��1�� ,X ,T� we have to put now the m+s conditions of orthogonality of

the discrepancy f�1�� ,X ,T� to the functions ��U�X,T��q� ��+�0�X ,T��

�0

2�

. . . �0

2�

��U�X,T��i�q� �� + �0�X,T��f �1�

i ��,X,T�dm�

�2��m = 0. �1.12�

The system �1.12� together with �1.8� gives m+ �m+s�=2m+s=N conditions at each X and Ton the parameters of zero approximation ��0�� ,X ,T� necessary for the construction of the first�-term in the solution �1.5�.

Let us prove now the following Lemma about the orthogonality conditions �1.12�:Lemma 1.1: Under all the assumptions of regularity formulated above the orthogonality

conditions (1.12) do not contain the functions �0��X ,T� and give just the restrictions on the

functions U��X ,T� having the form

C��q��U�UT

� − D��q��U�UX

� = 0 �1.13�

(with some functions C��q��U�, D�

�q��U�).

073505-3 Whitham systems and deformations J. Math. Phys. 47, 073505 �2006�


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Proof: Let us write down the part f�1�� of the function f�1� which contains the derivatives�0T

� �X ,T� and �0X� �X ,T�. We have from �1.11�

f �1��i ��,X,T� = −�Fi

��tj ��0�, . . . ��0��

j �0T� −

�Fi

��xj ��0�, . . . ��0��

j �0X�

−�Fi

��ttj ��0�, . . . �2��X,T��0��

j �0T� −

�Fi

��xxj ��0�, . . . �2k��X,T��0��

j �0X� − . . . .

Let us choose again the set of parameters U in the form

U = �k1, . . . ,km,�1, . . . ,�m,n1, . . . ,ns� ,

where k� are the wave numbers, �� are the frequencies of m-phase solutions, and �n1 , . . . ,ns� areadditional parameters �except the initial phases�.

We can write then

f �1��i ��,X,T� = −�

��Fi�� + �0,U�, . . . � + Lji �

�� j�� + �0,U��0T�

+ −�

�k�Fi�� + �0,U�, . . . � + Lji �

�k�� j�� + �0,U��0X� .

The derivatives �Fi /�� and �Fi /�k� are identically zero on according to �1.3�. We havethen

�0

2�

. . . �0

2�

��U�X,T��i�q� �� + �0�X,T��f �1��i ��,X,T�

dm�

�2��m 0

since all ��q�� ,X ,T� are the left eigenvectors of L with zero eigenvalues.It is easy to see also that all �0�X ,T� in the arguments of � and ��q� will disappear after the

integration with respect to � so we get the statement of the Lemma.Lemma 1.1 is proved.The system

�k�

�U�UT� =

��

�U� UX� , � = 1, . . . ,m ,

�1.14�C�

�q��U�UT� = D�

�q��U�UX� , q = 1, . . . ,m + s

is called the Whitham system for the m-phase solutions of system �1.1�.Let us note that we have rank ��k� /�U��=m according to our assumption above. In the generic

case the derivatives UT� can be expressed through UX

� and the Whitham system �1.14� can bewritten in the form

UT� = V�

� �U�UX�, �,� = 1, . . . ,N , �1.15�

where V�� U� is some N�N matrix depending on the variables U1 , . . . ,UN.

Let us say that quite often the system �1.1� can be written in the evolution form

�ti = Qi��,�x,�xx, . . . � . �1.16�

For systems �1.16� the form �1.15� of the corresponding Whitham system has then a naturalmotivation.

We will assume here that if the conditions �1.14� are satisfied then the system �1.9� is resolv-able on the space of 2�-periodic with respect to each �� function. The solution ��1�� ,X ,T� is



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defined then modulo a linear combination of the “right eigenfunctions” of L�X,T�ji ��0�� ,��0�nl�

introduced above. According to common approach4,12 we can try to use the corresponding coef-ficients to make the systems analogous to �1.9� resolvable in the next orders and try to findrecursively all the terms of series �1.5�.

Different aspects and numerous applications of the Whitham method were studied in manydifferent works1–51 �we apologize for the impossibility to give here the full list of numerous workson the Whitham method� and the Whitham method is considered now as one of the classicalmethods of investigation of nonlinear systems.

It was pointed out by G. Whitham1–3 that the Whitham system �1.14� has a local Lagrangianstructure in case when the initial system has a local Lagrangian structure

� � L��,�t,�x, . . . �dx dt = 0

on the space ��x , t��.The procedure of construction of Lagrangian formalism for the Whitham system �1.14� is

given by the averaging of the Lagrangian function L on the family of m-phase solutions of system�1.1�.1–3 Let us note also that in the case of the presence of additional parameters nl the additionalmethod of Whitham pseudophases should be used.

The important procedure of averaging of local field-theoretical Hamiltonian structures wassuggested by B. A. Dubrovin and S. P. Novikov.14,18,28,32 The Dubrovin-Novikov procedure givesthe local field-theoretical Hamiltonian formalism for the Whitham system �1.15� in the case whenthe initial system �1.16� has a local Hamiltonian formalism of general type. The Dubrovin-Novikov bracket for the Whitham system has a general form

U��X�,U��Y�� = g��U��X − Y� + b��U�UX

��X − Y� �1.17�

and was called the local Poisson bracket of hydrodynamic type. The theory of the brackets �1.17�is closely related with differential geometry14,28,32 and is connected with different coordinatesystems in the �pseudo� Euclidean spaces. Let us say also that during the last years the importantweakly nonlocal generalizations of Dubrovin-Novikov brackets �Mokhov-Ferapontov bracket andFerapontov brackets� were introduced and studied.53–59

The Hamiltonian structure �1.17� for the systems �1.15� has a direct relation to the integrabil-ity of the systems of this class. Thus it was conjectured by S. P. Novikov that any diagonalizablesystem �1.15� Hamiltonian with respect to the bracket of hydrodynamic type is integrable. Theconjecture of S. P. Novikov was proved by S. P. Tsarev,52 who suggested the “generalizedHodograph method” for solving the diagonal Hamiltonian systems �1.15�. Let us say that theTsarev method has become especially important for the Whitham systems corresponding to inte-grable hierarchies and provided a lot of very important solutions for such systems in differentcases. We note here that the Whitham systems of integrable hierarchies can usually be written indiagonal form3,11,25 and admit the �multi-� Hamiltonian structures given by the averaging of theLagrangian or the field-theoretical Hamiltonian structures of the initial system.

During the last years the theory of compatible Poisson brackets �1.17� and their deformationsin connection with Quantum Field Theory was intensively developed.60–72 Namely, the theory ofcompatible Poisson brackets �1.17� plays the main role in the theory of “Frobenius manifolds”constructed by B. A. Dubrovin and connected with the classification of the topological quantumfield theories �based on the Witten-Dijkgraaf-Verlinde-Verlinde equation�. Every “Frobenius mani-fold” is connected also with the integrable hierarchy of hydrodynamic type

Uts� = V�s��

� �U�UX� �1.18�

having the bi-Hamiltonian structure in Dubrovin-Novikov sense with a pair of Poisson brackets�1.17�. �The hierarchy �1.18� and the corresponding pair of Poisson brackets also possess someadditional properties �the existence of “unit vector field,” Euler field, �-symmetry, etc.��



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The “�-deformations” of the integrable hierarchies and the Poisson brackets �1.17� are con-nected with the “higher” corrections in the quantum field theories and are being intensivelyinvestigated now.64–72 According to B. A. Dubrovin and Y. Zhang the �-deformation of Frobeniusmanifold is given by the infinite series polynomial with respect to derivatives UX ,UXX ,UXXX , . . .and representing the “small dispersion” deformation of the hierarchy �1.18� and the correspondingbi-Hamiltonian structure of hydrodynamic type. The deformed hierarchy and bi-Hamiltonianstructure should possess also the additional properties of Frobenius manifolds �“unit vector field,”Euler field, �-symmetry etc.,¼� and give then the “deformation” of the corresponding Frobeniusmanifold.64,66,68 The general problem of classification of deformations of bi-Hamiltonian hierar-chies �1.18� is being also intensively studied by now and very important results were obtainedrecently in this area.67–72

The dispersive corrections to the Whitham systems were first considered by M. Y. Ablowitzand D. J. Benney �see Ref. 5, also Refs. 6 and 7� where the first consideration of multiphaseWhitham method was also made. As was shown in Ref. 5 the dispersive corrections to Whithamsystems can naturally arise and, besides that, can be generalized to the multiphase situation.

In this paper we will consider the deformations of the Whitham systems �1.15� havingDubrovin-Zhang form, i.e., the dispersive corrections to �1.15� containing the higher X-derivativesof the parameters U. The problem in this form is connected with the problem set by B. A.Dubrovin, which is formulated as the problem of deformations of Frobenius manifolds corre-sponding to the Whitham systems of integrable hierarchies. B. A. Dubrovin problem contains boththe problem of deformation of Whitham systems �1.15� and the corresponding bi-Hamiltonianstructures of hydrodynamic type giving the dispersive corrections to these structures.

We will consider here the first part of the B. A. Dubrovin problem and suggest a generalscheme of recursive construction of terms of the deformation of the Whitham system �1.15� usingthe initial system �1.1� or �1.16�. The second part of the B. A. Dubrovin problem will then beconsidered in the next paper. We will not require, however, the “integrability” of the system �1.15�and all the considerations below can be applicable for both integrable and nonintegrable systems�1.1� and �1.16�. Thus, we do not require also the bi-Hamiltonian property of the Whitham system�1.15�.

In the next section we will make more detailed investigation of the asymptotic series �1.5� anddescribe the construction of the deformation procedure for general Whitham systems.

II. THE DEFORMATION OF THE WHITHAM SYSTEMS

Let us start again with the description of the construction of asymptotic series �1.5� connectedwith the Whitham method. We will assume that we have the initial system having the general form�1.1� which has an N-parametric �modulo the initial phases �0

�� family of m-phase solutions

�i�x,t� = �i�k�U�x + ��U�t + �0,U� �2.1�

with the functions �� ,U� satisfying the system

Fi��,��U��,k��U��,��U��U��, . . . � = 0 �2.2�

�and 2�-periodic with respect to all ��.The choice of the functions �i�� ,U� at each U is defined modulo the initial phase �0 and the

full family of solutions of �2.2� is given by all the functions ��+�0 ,U� with arbitrary �0

= ��01 , . . . ,�0

m�. We will just assume here that the choice of �i�� ,U� is smooth on the family ofm-phase solutions .

We will assume also that the parameters �k ,�� are independent on the family such thatN 2m. It will be convenient to use the parameters �k1 , . . . ,km ,�1 , . . . ,�m ,n1 , . . . ,ns� on thefamily where k� are the “wave numbers,” �� are frequencies, and �n1 , . . . ,ns� are additionalparameters �if they present�. The linearly independent solutions of the linearized system �2.2� willthen be given by m+s functions �� ,U�, �nl�� ,U�.



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As we already said above we assume that the family represents the “full regular family” ofm-phase solutions of �2.3� such that the requirements formulated in Definition 1.1 or Definition1.1� are satisfied. We thus have exactly m+s functions �� ,U�, �nl�� ,U� giving the basis in thekernel of linearized system �2.2� and exactly m+s linearly independent functions ��U�

�q� ��+�0�giving the basis in the kernel of the adjoint operator �for generic k, �� and depending in a smoothway on the parameters �k ,� ,n�.

As we mentioned already we make a rescaling of coordinates X=�x, T=�t and try to find thesolutions of system

Fi��,��T,��X,�2�TT,�2�XT,�2�XX, . . . � = 0, i = 1, . . . ,n �2.3�

having the form

�i��,X,T,�� = �i�S�X,T��

+ �,X,T,�� , �2.4�

�i��,X,T,�� = �k 0

��k�i ��,X,T��k. �2.5�

The function ��0�� ,X ,T� belongs to the family at every fixed X and T,

��0�i ��,X,T� = �i�� + �0�X,T�,k�X,T�,��X,T�,n�X,T�� , �2.6�

and the compatibility conditions �1.12� for the system

Lji��1�

j ��,X,T� = f �1�i ��,X,T� �2.7�

give the Whitham system �1.13� on the parameters U�X ,T� of the zero approximation. The func-tion ��1�� ,X ,T� is defined from the system �2.7� modulo the linear combination of functions�� ,U�, �nl�� ,U� at every X and T.

All the other approximations ��k�� ,X ,T� satisfy the linear systems

Lji��k�

j ��,X,T� = f �k�i ��,X,T� , �2.8�

where the functions f�k�� ,X ,T� represent the higher order discrepancies given by system �2.3�.The compatibility conditions of the systems �2.8� �k 2� give the restrictions on the “initial

phases” �0� of the zero approximation and on the previous corrections ��k�� ,X ,T�.

Let us make now one more general assumption about the systems �2.7� and �2.8�. Namely weomit here the special investigation of the solvability of systems �2.7� and �2.8� on the space of2�-periodic functions and assume that the orthogonality conditions

�0

2�

. . . �0

2�

��U�X,T��i�q� �� + �0�X,T��f �k�


�2��m = 0 �2.9�

give the necessary and sufficient conditions of solvability of these systems. So, we will assumethat under the conditions �2.9� we can always find a smooth 2�-periodic solution of �2.7� and �2.8�defined modulo the linear combination

��=1

m

c�k�� X,T��,X,T� + �

l=1

s

d�k�l �X,T��nl��,X,T� . �2.10�

Let us consider now the construction of the asymptotic series �2.5�. We note first that thesolution ��k�� ,X ,T� is defined from the system �2.8� modulo the linear combination �2.10� whichis equivalent in the main order to the addition of the values �k+1c�k�

� �X ,T� to the phases S��X ,T� of



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the zero approximation ��0�� ,X ,T� and to the addition of the values �kd�k�l �X ,T� to the param-

eters nl�X ,T� of ��0�� ,X ,T� according to the formulas �2.4� and �2.5�. It is easy to see also thatwe can add the values �k+1c�k�X

� �X ,T� and �k+1c�k�T� �X ,T� to the parameters k� and �� in the

U-dependence of ��0�� ,X ,T� which does not affect the kth order of � in the series �2.5�.Let us change now the procedure of construction of series �2.5� in the following way:�1� At every step k we choose the solution ��k�� ,X ,T� in arbitrary way;�2� We allow the regular �-dependence

S��X,T,�� = �k 0

S�k�� X,T��k,

nl�X,T,�� = �k 0

n�k�l �X,T��k

of the phases S� and the parameters �k ,� ,n� of the zero approximation ��0�� ,X ,T� such that

k��X,T,�� = SX��X,T,��, ��X,T,�� = ST

��X,T,��

�3� We use the higher orders S�k�� , n�k�

l of the functions S��X ,T ,��, nl�X ,T ,�� to provide theorthogonality conditions �2.9� for the systems �2.8�.

We can now write the asymptotic solution of �2.3� in the form

�i��,X,T,�� = �i�S�X,T,��

+ �,SX�X,T,��,ST�X,T,��,n�X,T,�� + �k 1

��k�i �S�X,T,��

�+ �,X,T��k.

�2.11�

Let us note that the phase shift �0�X ,T� of the initial approximation now becomes the �-termin �-expansion of the function S�X ,T ,��.

It is not difficult to see that the series �2.11� can be always represented in the form �2.4� and�2.5� and give the same set of the asymptotic solutions of system �2.3�. However, the series �2.11�contains a big “renormalization freedom” since we can change in arbitrary way the higher orders

corrections S�2��X ,T�, S�3��X ,T� , . . ., n�1��X ,T�, n�2��X ,T� , . . . and then adjust the functions ��k� inthe appropriate way.

Let us now “fix the normalization” of the solution �2.11� in the following way.We require that the first term

�i�S�X,T,��

+ �,SX�X,T,��,ST�X,T,��,n�X,T,��of �2.11� gives “the best approximation” of the asymptotic solution �2.11� by the modulatedm-phase solution of �1.1�. To be more precise, we require that the rest of the series �2.11� isorthogonal �at every X and T� to the vectors �� ,X ,T�, �nl�� ,X ,T� “tangent” to at the“point” �S�X ,T ,�� ,n�X ,T ,��.

Thus we will now put the conditions

�0

2�

. . . �0

2�

�i=1

n

��i ��,SX,ST,n��

k=1

�

��k�i ��,X,T��k� dm�

�2��m = 0, �2.12�

�0

2�

. . . �0

2�

�i=1

n

�nli ��,SX,ST,n��

k=1

�

��k�i ��,X,T��k� dm�

�2��m = 0 �2.13�

�for all ��.



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The representation �2.6� defines in fact the first correction S�1�=��0��X ,T�. As we saw above,

the functions ��k�� ,X ,T� are defined modulo the linear combinations of the functions�� ,S�0�X ,S�0�T ,n�0��, �nl�� ,S�0�X ,S�0�T ,n�0�� which are linearly independent for the full regularfamily of m-phase solutions of �1.1�. Using this fact it is not difficult to prove that the conditions

�2.12� and �2.13� fix uniquely all the terms S�k+1�, n�k�, and ��k�� ,X ,T� �k 1� for a givensolution �2.4� and �2.5� which gives then the representation �2.11�.

Let us say now that the choice of the normalization �2.12� and �2.13� is not unique. Inparticular, it depends on the choice of the variables �i�x , t� for the “vector” �i�1� systems �1.1�.We will speak about the normalization �2.12� and �2.13� as about one possible way to fix the

functions ��k�� ,X ,T�.The solution �� ,X ,T� can now be defined as the asymptotic solution of the system �2.3�

having the form �2.11� which satisfies the conditions

�0

2�

. . . �0

2�

�i=1

n

��i ��,SX,ST,n��i�� −

S

�,X,T,�� dm�

�2��m = 0,

�0

2�

. . . �0

2�

�i=1

n

�nli ��,SX,ST,n��i�� −

S

�,X,T,�� − �i��,SX,ST,n�� dm�

�2��m = 0

�for all ��.To find a solution in the form �2.11� we substitute the series �2.11� in the system �2.3� and try

to find the functions ��k�i �� ,X ,T� from the linear systems

Lj�S�0�,n�0��i ��k�

j ��,X,T� = f �k�i ��,X,T� �2.14�

analogous to �2.8�.The functions f �k�

i �� ,X ,T� are different from the functions f �k�i �� ,X ,T� since we included “a

part of �-dependence” in the first term of �2.11�. The functions f�k�� ,X ,T� should be orthogonalto the functions ��S�0�,n�0��

�q� �� ,X ,T�, q=1, . . . ,m+s and we use the conditions �2.12� and �2.13� for

the recurrent determination of ��k�� ,X ,T�. The functions S�k�� X ,T�, n�k�

l �X ,T� are used now toprovide the resolvability of the systems �2.14� in all orders of �.

Let us investigate now the systems arising on the functions S�0��X ,T�, S�1��X ,T� , . . .,n�0��X ,T�, n�1��X ,T� , . . .. Let us note that we have in our notations S�X ,T�=S�0��X ,T�, �0�X ,T�=S�1��X ,T�. We saw in Lemma 1.1 that the function �0�X ,T� does not appear in the solvabilityconditions of the system �1.9�. For the asymptotic solution written in the form �2.11� with thenormalization conditions �2.12� and �2.13� we can prove here even stronger statement.

Lemma 2.1: The functions S�k�� X ,T�, n�k�

l �X ,T� do not appear in the expression for the dis-

crepancy f�k�� ,X ,T� and do not affect the solution ��k�� ,X ,T�.Proof: The way we get the discrepancy f�k�� ,X ,T� can be described as follows:�1� We substitute the solution �2.11� in the system �2.3�;�2� After “making all differentiations” we can omit the argument shift S�X ,T ,�� /� in all

functions depending on �;�3� Then collecting together all the terms of order �k we get the system �2.14�.It is not difficult to see that in this approach the kth order of � containing the functions

S�k��X ,T� or n�k��X ,T� have the form



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dFi

dk�S�k�X� ,

dFi

d��S�k�T� dFi

dnl n�k�l ,

where Fi are the constraints �2.2� defining the m-phase solutions of �1.1�. The derivatives d/dk�,d /d��, d /dnl here are the total derivatives including the dependence of the functions�i�� ,k ,� ,n� on the corresponding parameters view the form of �2.11�. However, all these de-rivatives are identically equal to zero on the family , so we get the first part of the Lemma. Tofinish the proof we have to note also that the values S�k��X ,T�, n�k��X ,T� do not appear in the kthorder of � of the normalization conditions �2.12� and �2.13� either.

Lemma 2.1 is proved.We can now formulate the procedure in the following form:We try to solve the systems �2.14� recursively in all orders and find the functions

��k�� ,X ,T� satisfying the conditions �2.12� and �2.13�. At each kth step of our procedure we geta system on the functions S�k−1��X ,T�, n�k−1��X ,T� from the solvability conditions of �2.14�. Thefull set of Whitham solutions will then be parametrized by the set of all functions S�k��X ,T� ,n�k��X ,T��, k 0 satisfying the conditions of solvability of systems �2.14�.

Now let us investigate the systems arising on the functions S�k−1��X ,T�, n�k−1��X ,T� from thesolvability conditions of �2.14�

�0

2�

. . . �0

2�

��S�0�,n�0��i�q� ��,X,T� f �k�


�2��m 0, q = 1, . . . ,m + s �2.15�

in the kth order of �. We note first that the solvability conditions of �2.14� for k=1 give a nonlinearsystem on the functions S�0��X ,T�, n�0��X ,T�. It is not difficult to prove the following Lemma:

Lemma 2.2: The system arising for k=1 coincides with the Whitham system for the functionsS�0��X ,T�, n�0��X ,T�.

Proof: Indeed, using Lemma 2.1 it is not difficult to see that the discrepancy f�1�� ,X ,T�differs from f�1�� ,X ,T� just by the terms containing derivatives �0X and �0T. However, accordingto Lemma 1.1 these terms do not affect the orthogonality conditions �1.12� which then coincide

with orthogonality conditions for f�1�� ,X ,T�.Lemma 2.2 is proved.The Whitham system �orthogonality conditions� for the functions S�0��X ,T�, n�0��X ,T� can be

written in the following general form:

W�q��S�0�,n�0��X,T� ��S�0�,n�0��q� · f�1��S�0�,n�0��X,T�

= �0

2�

. . . �0

2�

��S�0�,n�0��i�q� ��,X,T� f �1�

i �S�0�,n�0��,X,T�dm�

�2��m

= A��q��S�0�X,S�0�T,n�0��S�0�TT

�

+ B��q��S�0�X,S�0�T,n�0��S�0�XT

� + C��q��S�0�X,S�0�T,n�0��S�0�XX

�

+ Gl�q��S�0�X,S�0�T,n�0��n�0�T

l + Hl�q��S�0�X,S�0�T,n�0��n�0�X

l = 0 �2.16�

with some functions A��q�, B�

�q�, C��q�, Gl

�q�, Hl�q�, q=1, . . . ,m+s, �=1, . . . ,m, l=1, . . . ,s.

Let us prove now the following Lemma about the systems on S�k�, n�k�, k 1.

Lemma 2.3: The orthogonality conditions of f�k+1�� ,X ,T� to the functions ��S�0�,n�0��q� give the

following linear systems for the functions S�k��X ,T�, n�k��X ,T�, k 1:



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� � W�q��X,T�S�0�

� �X�,T��S�k�

� �X�,T��dX�dT� +� � W�q��X,T�n�0�

l �X�,T��n�k�

l �X�,T��dX�dT�

= V�k��q��S�0�, . . . ,S�k−1�,n�0�, . . . ,n�k−1��X,T� ,

where

W�q��X,T�S�0�

� �X�,T��=

�W�q�

�S�0�T� ��T − T��X − X�� +

�W�q�

�S�0�X� �T − T��X − X��

+�W�q�

�S�0�TT� ��T − T��X − X�� +

�W�q�

�S�0�XT� ��T − T��X − X�� +

�W�q�

�S�0�XX� �T − T�� X − X�� ,

W�q��X,T�n�0�

l �X�,T��=

�W�q�

�n�0�l �T − T��X − X�� +

�W�q�

�n�0�Tl ��T − T��X − X�� +

�W�q�

�n�0�Xl �T − T��X − X�� .

In other words, the functions S�k��X ,T�, n�k��X ,T� satisfy the linearized Whitham system onthe functions S�0��X ,T�, n�0��X ,T� with additional right-hand part depending on the functionsS�0� , . . ., S�k−1�, n�0� , . . ., n�k−1�.

Proof: Let us look at the terms in f�k+1�� ,X ,T� which contain the functions S�k��X ,T�,n�k��X ,T�:

�1� As we proved in Lemma 2.1 the functions f �1�i �� ,X ,T� contain only the terms depending

on S�0�, n�0�. It is easy to see that the functions f �k+1�i �� ,X ,T� will then contain the terms

� � f �1�i ��,X,T�

S�0�� X�,T��

S�k�� X�,T��dX� dT� +� � f �1�

i ��,X,T�

n�0�l �X�,T��

n�k�l �X�,T��dX� dT�

according to the form of the first term in �2.11�.�2� There are terms containing the functions S�k�, n�k� and the function ��1�� ,X ,T�. All such

terms can be written in the form:

−� � S�k�� X�,T��

L�S�0�,n�0��ji �X,T�

S�0�� X�,T��

+ n�k�l �X�,T��

L�S�0�,n�0��ji �X,T�

n�0�l �X�,T��

��1�j ��,X,T�dX� dT�,

where Lj�S�0�,n�0��i is the linear operator �1.10� given by the linearization of the system �2.2� on the

family of m-phase solutions.�3� There are terms of the form

−� . . .� 2F�S�0�,n�0��i ��,X,T�

S�0�� X�,T��S�0�

� �X�,T ��S�k�

� �X�,T��S�1�� X�,T ��dX� dT� dX� dT � − ¯ ,

where F�S�0�,n�0��i �� ,X ,T� is the left-hand part of the system �2.2�.

However, the sum of all such terms is equal to zero since they all correspond to the expansionof the “shift of parameters” S�0��X ,T�, n�0��X ,T� on the family where we have Fi�� ,X ,T�0identically.

Let us look now at the terms in the orthogonality conditions

�0

2�

. . . �0

2�

��S�0�,n�0��i�q� ��,X,T� f �k+1�


�2��m = 0 �2.17�

containing the terms �1� and �2�.



131.230.73.202 On: Fri, 19 Dec 2014 01:33:46

We have identically

�0

2�

. . . �0

2�

��S�0�,n�0��i�q� ��,X,T�L�S�0�,n�0��j

i ��,��,X,T�dm�

�2��m 0 �2.18�

on �where L�S�0�,n�0��ji �� ,�� ,X ,T� is the “core” of the operator L�S�0�,n�0��j

i �X ,T��.It is easy to see then that the inner product of ��q� with the terms �2� is equal to

�0

2�

. . . �0

2� � � S�k�� X�,T��

��S�0�,n�0��i�q� ��,X,T�

S�0�� X�,T��

+ n�k�l �X�,T��

��S�0�,n�0��i�q� ��,X,T�

n�0�l �X�,T��

�dX� dT�

�L�S�0�,n�0��ji �X,T��1�

j ��,X,T�dm�

�2��m .

It is not difficult to see now that the terms of orthogonality conditions containing the terms �1�and �2� can be written together in the form

� � S�k�� X�,T��

S�0�� X�,T��

��S�0�,n�0��q� · f�1��S�0�,n�0��X,T�dX� dT�

+� � n�k�l �X�,T��

n�0�l �X�,T��

��S�0�,n�0��q� · f�1��S�0�,n�0��X,T�dX� dT�

�where �¯¯� is the inner product of ��q� and f�1��.All the other terms in the orthogonality conditions �2.17� are the smooth functionals of

S�0� , . . . ,S�k−1�, n�0� , . . . ,n�k−1�, so we get the statement of the Lemma.Lemma 2.3 is proved.Let us consider now the systems �1.1� satisfying the special nondegeneracy conditions.

Namely, we will assume that the corresponding Whitham system �2.16� can be resolved withrespect to the highest T derivatives of the functions S�0��X ,T�, n�0��X ,T� and written in the “evo-lution” form:

S�0�TT� = M�0��

� �S�0�X,S�0�T,n�0��S�0�XX� + N�0��

� �S�0�X,S�0�T,n�0��S�0�TX� + P�0�p

� �S�0�X,S�0�T,n�0��n�0�Xp ,

�2.19�� = 1, . . . ,m ,

n�0�Tl = T�0��

l �S�0�X,S�0�T,n�0��S�0�XX� + L�0��

l �S�0�X,S�0�T,n�0��S�0�TX� + R�0�p

l �S�0�X,S�0�T,n�0��n�0�Xp ,

�2.20�l = 1, . . . ,s .

After the introduction of the variables k�0�� =S�0�X

� , ��0�� =S�0�T

� we can write the system �2.19�and �2.20� in the form

k�0�T� = ��0�X

� ,

��0�T� = M�0��

� �k�0�,��0�,n�0��k�0�X� + N�0��

� �k�0�,��0�,n�0��0�X� + P�0�p

� �k�0�,��0�,n�0��n�0�Xp ,

n�0�Tl = T�0��

l �k�0�,��0�,n�0��k�0�X� + L�0��

l �k�0�,��0�,n�0��0�X� + R�0�p

l �k�0�,��0�,n�0��n�0�Xp ,

�2.21�

i.e., in the form �1.15�.We will consider now the hyperbolic systems �2.21�, i.e., such that the matrix



131.230.73.202 On: Fri, 19 Dec 2014 01:33:46

V�� k�0�,��0�,n�0�� = � 0 Im 0

M�0� N�0� P�0�

T�0� L�0� R�0��

has exactly N=2m+s real eigenvalues with N linearly independent real eigenvectors. For hyper-bolic systems �2.21� it is natural to consider the Cauchy problem with the smooth initial datak�0��X ,0�, ��0��X ,0�, n�0��X ,0� �or S�0��X ,0�, S�0�T�X ,0�, n�0��X ,0��. The smooth solution of �2.21�exists in general up to some finite time T0 until the breakdown occurs. So we can write the zero�global in X� approximation for the solution �2.11� just in the time interval where we have asmooth solution of the Whitham system. Using Lemma 2.3 it is not difficult to prove then thefollowing Lemma:

Lemma 2.4: For nondegenerate hyperbolic Whitham system (2.21) and the global solutionS�0��X ,T�, n�0��X ,T� defined on the interval �0,T0� the higher orders approximations in (2.11) areall defined for all X and T� �0,T0� and are parametrized by the initial values S�k��X ,0�,S�k�T�X ,0�, n�k��X ,0�. �Let us recall that we assume that all systems �2.14� are solvable if thecorresponding orthogonality conditions are satisfied.�

Proof: Indeed, as follows from Lemma 2.3 the functions S�k��X ,T�, n�k��X ,T� are defined bythe initial values S�k��X ,0�, S�k�T�X ,0�, n�k��X ,0� and can be found from the linear system using thecharacteristic directions of �2.21� �defined by S�0��X ,T�, n�0��X ,T�� provided that all the smooth

solutions S�0��X ,T� , . . . ,S�k−1��X ,T�, n�0��X ,T� . . . ,n�k−1��X ,T� and ��1�� ,X ,T� , . . . ,

��k−2�� ,X ,T� exist on the interval �0,T0�. According to Lemma 2.1 and Lemma 2.3 we can find

then the functions ��k−1�i �� ,X ,T� which are the local expressions �in X and T� of

S�0��X ,T� , . . . ,S�k−1��X ,T�, n�0��X ,T� . . . ,n�k−1��X ,T� and their derivatives. Using the induction wethen finish the proof of the Lemma.

Lemma 2.4 is proved.According to Lemma 2.4 we can formulate now the following statement:For the initial system �1.1� having the nondegenerate hyperbolic Whitham system �2.21� the

corresponding Whitham solutions �2.11� �or �2.4� and �2.5�� are defined by the initial valuesS�X ,0 ,��, ST�X ,0 ,��, n�X ,0 ,�� and exist in the time interval �0,T0� defined by the Whithamsystem �2.21� and the initial data S�0��X ,0�=S�X ,0 ,0�, S�0�T�X ,0�=ST�X ,0 ,0�, and n�0��X ,0�=n�X ,0 ,0�.

The deformation procedure: Let us note now that the series �2.11� �or �2.4� and �2.5�� give infact the one-parametric formal solutions of �1.1� with a parameter �. Let us rewrite now thesolutions �2.11� in the form which gives the concrete �formal� solution of �1.1� and is not con-nected with the additional one-parametric �-family including this given solution. We omit now the�-dependence of functions S�X ,T ,��, n�X ,T ,�� or put formally �=1� and say that the Whithamsolution is defined now by functions S�X ,T�, n�X ,T� determined by the initial values S�X ,0�,ST�X ,0�, and n�X ,0�. �Let us keep here the notations X and T for the spatial and time coordinatesjust to emphasize that we consider the “slow” functions SX�X ,T�, ST�X ,T�, n�X ,T�.�

Thus, we define now the Whitham solution as the solution of the system

Fi��,�T,�X,�TT,�XT,�XX, . . . � = 0, i = 1, . . . ,n �2.22�

having the form

�i��,X,T� = �i�S�X,T� + �,SX�X,T�,ST�X,T�,n�X,T�� + �k 1

��k�i �S�X,T� + �,X,T�

�2.23�

where the functions ��k�i :

�1� Are 2�-periodic with respect to each ��;�2� Have degree k �introduced below�;



131.230.73.202 On: Fri, 19 Dec 2014 01:33:46

�3� Satisfy the normalization conditions �2.12� and �2.13� which will be written now in theform:

�0

2�

. . . �0

2�

�i=1

n

��i ��,SX,ST,n��k�


�2��m = 0, �2.24�

�0

2�

. . . �0

2�

�i=1

n

�nli ��,SX,ST,n��k�


�2��m = 0 �2.25�

k 1, ��=1, . . . ,m, l=1, . . . ,s�.Let us introduce now the gradation used for the formal expansion �2.23�. Namely, for the

systems �1.1� having the nondegenerate hyperbolic Whitham system �2.21� we put the followinggradation on the functions SX�X ,T�, ST�X ,T�, n�X ,T� and their derivatives:

�1� The functions k��X ,T�=SX��X ,T�, ��X ,T�=ST

��X ,T�, and nl�X ,T� have degree 0;�2� Every differentiation with respect to X adds 1 to the degree of the function;�3� The degree of the product of two functions having certain degrees is equal to the sum of

their degrees.In other words, for the parameters U= �k ,� ,n� we have the gradation rule of Dubrovin-Zhang

type, i.e.,All the functions f�U� have degree 0;The derivatives UkX

� have degree k;The degree of the product of functions having certain degrees is equal to the sum of their

degrees.We put now the evolution conditions to the functions S�X ,T�, n�X ,T� having the form:

STT� = �

k 1��k�

� �k,�,n,kX,�X,nX, . . . � , �2.26�

nTl = �

k 1��k�

l �k,�,n,kX,�X,nX, . . . � , �2.27�

where ��k�� , ��k�

l are general polynomials in derivatives kX ,�X ,nX ,kXX ,�XX ,nXX , . . . �with coeffi-cients depending on �k ,� ,n�� having degree k.

We now substitute the expansion �2.23� in the system �2.22� and use the relations �2.26� and�2.27� to remove all the time derivatives of parameters �k ,� ,n�. After that we can divide thesystem �2.22� into the terms of certain degrees and try to find recursively all the terms��k�� ,X ,T� for k=1,2 , . . ..

It is easy to see again that for any ��k�� ,X ,T� we will have the linear system analogous to�2.8� and �2.14�, i.e.,

L�S�X,T�,n�X,T��ji ��k�

i ��,X,T� = f �k�i ��,X,T� , �2.28�

where f�k�� ,X ,T� is the discrepancy having degree k according to the definition above.We have to put again the orthogonality conditions

�0

2�

. . . �0

2�

��S,n�i�q� ��,X,T� f �k�


�2��m 0 �2.29�

on the functions f �k�i �� ,X ,T� and then find the unique ��k�� ,X ,T� satisfying the normalization

conditions �2.24� and �2.25�.It is not difficult to prove the following Lemma:Lemma 2.5: (1) For any system (1.1) having the nondegenerate hyperbolic Whitham system



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(2.21) the functions ��k�� , ��k�

l are uniquely determined by the orthogonality conditions (2.29) in theorder k.

(2) The functions ��1�� , ��1�

l give the Whitham system (2.19) and (2.20) for the functionsS�0��X ,T�, n�0��X ,T�. �In fact, the functions S�X ,T ,��, n�X ,T ,�� introduced previously satisfy thefull system �2.26� and �2.27�.�

Proof: Indeed, using Lemma 2.1 it is easy to see that the functions f �1�i �� ,X ,T� coincide with

the functions f �1�i �� ,X ,T� introduced in �2.14� after the replacement of functions S�0��X ,T�,

n�0��X ,T� by S�X ,T�, n�X ,T�. Comparing then the orthogonality conditions �2.29� with �2.15� weget the second part of the Lemma.

To prove the first part we just note that all ��k�� , ��k�

l arise in the kth order of system �2.22� “inthe same way.” We can conclude then that the orthogonality conditions �2.29� in the kth orderalways contain the functions ��k�

� , ��k�l in one particular way which coincides with the appearance

of ��1�� , ��1�

l in the Whitham system arising for k=1. From the definition of the nondegenerateWhitham system we now obtain the first part of the Lemma.

Lemma 2.5 is proved.Definition 2.1: We call the system (2.26) and (2.27) or the equivalent system

kT� = �X

�,

�T� = �

k 1��k�

� �k,�,n,kX,�X,nX, . . . � ,

nTl = �

k 1��k�

l �k,�,n,kX,�X,nX, . . . � �2.30�

the deformation of the Whitham system (2.19) and (2.20) (or (2.21)).The functions k��X ,T�, ��X ,T�, nl�X ,T� are the “slow” functions of the variables x and t and

the system �2.30� gives the analog of the “low-dispersion” expansion in our case. The asymptoticsolutions �2.23� of the initial system �2.22� are parametrized by the asymptotic solutions k��X ,T�,��X ,T�, nl�X ,T� of the system �2.30� and arbitrary �constant� initial phases �0

�. As follows fromour considerations above the solutions �2.23� give all the “particular solutions” �2.4� and �2.5�,however, they do not contain the additional information about the one-parametric �-family givenby �2.4� and �2.5�

Remark 1: Let us note that the full set of parameters of m-phase solutions of �1.1� is given byk, �, n, and �0. However the functions �0�X ,T� do not present as the parameters of solutions�2.23� in this approach. This shows in fact that the introduction of the functions �0

��X ,T� does notgive “new” formal solutions of �1.1� and is responsible for the additional �-dependence of one-parametric families �2.4� and �2.5�. Here they are “absorbed” by the total phase S�X ,T� connectedwith the “particular” formal solution of �1.1�.

Remark 2: In our consideration we fixed some functions �i�� ,k ,� ,n� on the family �foreach �k ,� ,n�� as having zero initial phases. However, the choice of the functions �i�� ,k ,� ,n�is not unique. In particular, the natural change of the functions �i�� ,k ,� ,n� on can be writtenin the form

��,k,�,n� = �� + �0�k,�,n�,k,�,n� , �2.31�

where �0��k ,� ,n� are arbitrary smooth functions.

It is not difficult to see also that the system �2.30� depends on the choice of the functions�i�� ,k ,� ,n� in the high �k 2� orders.

Let us give here the definition given by B. A. Dubrovin and Y. Zhang64,66 and connected withthe “equivalence” of different infinite �or finite� systems.

Definition 2.2 �B. A. Dubrovin, Y. Zhang�: Consider the system of the form



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UT� = �

k 1V�k�

� �U,UX,UXX, . . . �, � = 1, . . . ,N �2.32�

for arbitrary parameters U� where all V�k�� are smooth functions polynomial in UX ,UXX , . . . and

having degree k. We say that two systems (2.32) are connected by the triviality transformation (orequivalent) if they are connected by the formal substitution

U� = �k 0

U�k�� U,UX,UXX, . . . � ,

where all U�k�� are smooth functions polynomial in UX ,UXX , . . . and having degree k.

Let us say actually that the definition of B. A. Dubrovin and Y. Zhang is applied usually to thewhole integrable hierarchies and plays the important role in the classification of integrable hier-archies having the form �2.32�. We will prove here the following Lemma:

Lemma 2.6: The deformations of the Whitham system (2.30) written for the initial functions�� ,k ,� ,n� and �� ,k ,� ,n� connected by the transformation (2.31) are equivalent inDubrovin-Zhang sense.

Proof: Let us prove first the following statement:For any transformation �2.31� there exists a unique change of functions S��X�, nl�X�:

S�� = S� − �0��k,�,n� + �

k 1S�k�

� �k,�,n,kX, . . . � ,

n�l = nl + �k 1

n�k�l �k,�,n,kX, . . . � �2.33�

such that:�1� All S�k�, n�k� are polynomial in derivatives of �k ,� ,n� and have degree k;�2� For any solution �2.23� ��S,n�� ,X ,T�� of �2.22� the functions

��i�S��X ,T�+� ,SX� ,ST� ,n�� satisfy the normalization conditions

�0

2�

. . . �0

2�

�i=1

n

��i �S� + �,SX� ,ST�,n��S,n�


�2��m 0, �2.34�

�0

2�

. . . �0

2�

�i=1

n

�n�l�i �S� + �,SX� ,ST�,n��S,n�i ��,X,T� − ��i�S� + �,SX� ,ST�,n��

dm�

�2��m 0.

�2.35�

For the proof of this statement let us note first that we can always express all the timederivatives of k, �, and n using the system �2.30� in terms of X-derivatives of these functions.Using this procedure we try to find the transformation �2.33� recursively in all degrees by substi-tution of �2.33� in �2.34� and �2.35�. It is not difficult to check then that the functions S�k�

� , n�k�l are

defined in the kth order of �2.34� and �2.35� from a linear system. The matrix of this linear systemcoincides with the Gram matrix of functions ��, �nl at every X and T. Thus, for the fullnondegenerate family of m-phase solutions of �1.1� this system has a unique solution at everydegree k. The transformation �2.33� is evidently invertible in sense of the infinite series �polyno-mial with respect to derivatives of �k ,� ,n�� so we can also express the functions �S ,n� in termsof �S� ,n��.

We can now try to use the functions ��i�S�+� ,SX� ,ST� ,n�� as the zero approximation in the�S� ,n��-expansion of the corresponding solution �2.23�. It is not difficult to see that the difference



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��i�S� + �,SX� ,ST�,n�� − ��S,n�i ��,X,T�

can be represented as the infinite series polynomial with respect to derivatives of �k ,� ,n� andstarting with the terms of degree 1. After the expression of the functions �k ,� ,n� in terms of�k� ,�� ,n�� in this difference we get finally the �S� ,n��-expansion of the solution ��S,n�

i �� ,X ,T�.The functions �S� ,n�� now satisfy the new deformed Whitham system �2.26� and �2.27�

corresponding to the choice of the functions �� ,k� ,�� ,n�� as the functions of zero approxi-mation.

It is easy to see also that the transformation �2.33� remains polynomial in X-derivatives of�k ,� ,n� after the expression of time derivatives of �k ,� ,n� using the system �2.30�.

We obtain then that the transformation �2.33� gives a “triviality” connection between thesystems �2.30� written for the initial functions �� ,k ,� ,n� and �� ,k ,� ,n�.

Lemma 2.6 is proved.At the end let us note again that Lemma 2.6 is important in fact for the integrable hierarchies

rather than for the one particular system �1.1� according to Dubrovin-Zhang approach to classifi-cation of integrable systems.

ACKNOWLEDGMENT

I wish to thank Professor B. A. Dubrovin, who suggested the problem, for the interest in thiswork and many fruitful discussions.

1 G. Whitham, J. Fluid Mech. 22, 273 �1965�.2 G. Whitham, Proc. R. Soc. London, Ser. A 139, 283 �1965�.3 G. Whitham, Linear and Nonlinear Waves �Wiley, New York, 1974�.4 J. C. Luke, Proc. R. Soc. London, Ser. A 292, 403 �1966�.5 M. J. Ablowitz and D. J. Benney, Stud. Appl. Math. 49, 225 �1970�.6 M. J. Ablowitz, Stud. Appl. Math. 50, 329 �1971�.7 M. J. Ablowitz, Stud. Appl. Math. 51, 17 �1972�.8 W. D. Hayes, Proc. R. Soc. London, Ser. A 332, 199 �1973�.9 A. V. Gurevich and L. P. Pitaevskii, JETP Lett. 17, 193 �1973�.

10 A. V. Gurevich and L. P. Pitaevskii, Sov. Phys. JETP 38, 291 �1974�.11 H. Flaschka, M. G. Forest, and D. W. McLaughlin, Commun. Pure Appl. Math. 33, 739 �1980�.12 S. Yu. Dobrokhotov and V. P. Maslov, Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fundam. Napravleniya 15, 3 �1980�

�in Russian�; J. Sov. Math. 15, 1433 �1980� �English translation�.13 V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Teoriya Solitonov Metod Obratnoi Zadachi, edited

by S. P. Novikov �Nauka, Moscow 1980� �in Russian�; S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E.Zakharov, Theory of Solitons. The Inverse Scattering Method �Plenum, New York 1984� �English translation�.

14 B. A. Dubrovin and S. P. Novikov, Sov. Math. Dokl. 27, 665 �1983�.15 P. D. Lax and C. D. Levermore, Commun. Pure Appl. Math. 36, 253 �1983�; 36, 571 �1983�; 36, 809 �1983�.16 S. Venakides, Commun. Pure Appl. Math. 38, 125 �1985�.17 S. Venakides, Commun. Pure Appl. Math. 38, 883 �1985�.18 S. P. Novikov, Russ. Math. Surveys 40, 85 �1985�.19 V. V. Avilov and S. P. Novikov, Sov. Phys. Dokl. 32, 366 �1987�.20 A. V. Gurevich and L. P. Pitaevskii, Sov. Phys. JETP 66, 490 �1987�.21 I. M. Krichever and S. P. Novikov, Sov. Phys. Dokl. 32, 564 �1987�.22 S. Venakides, Trans. Am. Math. Soc. 301, 189 �1987�.23 M. V. Pavlov, Theor. Math. Phys. 71, 584 �1987�.24 C. D. Levermore, Commun. Partial Differ. Equ. 13, 495 �1988�.25 I. M. Krichever, Funct. Anal. Appl. 22, 200 �1988�.26 G. V. Potemin, Russ. Math. Surveys 43, 252 �1988�.27 R. Haberman, Stud. Appl. Math. 78�1�, 73 �1988�.28 B. A. Dubrovin and S. P. Novikov, Russ. Math. Surveys 44, 35 �1989�.29 A. V. Gurevich, A. L. Krylov, and G. A. El, JETP Lett. 54, 102 �1991�.30 A. V. Gurevich, A. L. Krylov, and G. A. El, Sov. Phys. JETP 74, 957 �1992�.31 I. M. Krichever, Sov. Sci. Rev., Sect. C, Math. Phys. Rev. 9, 1 �1992�.32 B. A. Dubrovin and S. P. Novikov, Sov. Sci. Rev., Sect. C, Math. Phys. Rev. 9, 1 �1993�.33 P. D. Lax, C. D. Levermore, and S. Venakides, Important Developments in Soliton Theory 1980–1990, Springer Series

in Nonlinear Dynamics �Springer, Berlin, 1993�, pp. 205–241.34 F.-R. Tian, Commun. Pure Appl. Math. 46, 1093 �1993�.35 F.-R. Tian, Commun. Math. Phys. 166, 79 �1994�.36 I. M. Krichever, Commun. Pure Appl. Math. 47, 437 �1994�.



131.230.73.202 On: Fri, 19 Dec 2014 01:33:46

37 A. Ya. Maltsev and M. V. Pavlov, Funct. Anal. Appl. 29, 6 �1995�.38 M. V. Pavlov, Dokl. Math. 50, 220 �1995�.39 B. A. Dubrovin, Am. Math. Soc. Transl. 179, 35 �1997�.40 A. Ya. Maltsev, Izvestiya, Mathematics 63, 1171 �1999�.41 T. Grava, Russ. Math. Surveys 54, 169 �1999�.42 T. Grava, Theor. Math. Phys. 122, 46 �2000�.43 T. Grava, Math. Phys., Anal. Geom. 4, 65 �2001�.44 G. A. El, A. L. Krylov, and S. Venakides, Commun. Pure Appl. Math. 54, 1243 �2001�.45 A. Ya. Maltsev, Int. J. Math. Math. Sci. 30, 399 �2002�.46 T. Grava, Commun. Pure Appl. Math. 55, 395 �2002�.47 T. Grava and F.-R. Tian, Commun. Pure Appl. Math. 55, 1569 �2002�.48 G. A. El, Phys. Lett. A 311, 374 �2003�.49 T. Grava, Acta Appl. Math. 82, 1 �2004�.50 A. Ya. Maltsev, J. Phys. A 38, 637 �2005�.51 S. Abenda and T. Grava, Ann. Inst. Fourier 55, 1001 �2005�.52 S. P. Tsarev, Sov. Math. Dokl. 31, 488 �1985�.53 O. I. Mokhov and E. V. Ferapontov, Russ. Math. Surveys 45, 218 �1990�.54 E. V. Ferapontov, Funct. Anal. Appl. 25, 195 �1991�.55 E. V. Ferapontov, Funct. Anal. Appl. 26, 298 �1992�.56 E. V. Ferapontov, Theor. Math. Phys. 91, 642 �1992�.57 E. V. Ferapontov, Am. Math. Soc. Transl. 170, 33 �1995�.58 M. V. Pavlov, Dokl. Math. 59, 374 �1995�.59 A. Ya. Maltsev and S. P. Novikov, Physica D 156, 53 �2001�.60 B. A. Dubrovin, Nucl. Phys. 379, 627 �1992�.61 B. A. Dubrovin, ArXiv: hep-th/9209040.62 B. A. Dubrovin, ArXiv: hep-th/9407018.63 B. A. Dubrovin, in Proceedings of the 1997 Taniguchi Symposium Integrable Systems and Algebraic Geometry, edited by

M.-H. Saito, Y. Shimizu, and K. Ueno �World Scientific, Singapore, 1998�, pp. 47–72.64 B. A. Dubrovin and Y. Zhang, Commun. Math. Phys. 198, 311 �1998�.65 B. A. Dubrovin, ArXiv: math.AG/9807034.66 B. A. Dubrovin and Y. Zhang, ArXiv: math.DG/0108160.67 P. Lorenzoni, J. Geom. Phys. 44, 331 �2002�.68 B. A. Dubrovin and Y. Zhang, ArXiv: math.DG/0308152.69 S.-Q. Liu and Y. Zhang, ArXiv: math.DG/0405146.70 S.-Q. Liu and Y. Zhang, ArXiv: math.DG/0406626.71 B. Dubrovin, S.-Q. Liu, and Y. Zhang, ArXiv: math.DG/0410027.72 B. Dubrovin, Y. Zhang, and D. Zuo, ArXiv: math.DG/0502365.



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whitham systems and deformations

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