On the classification of Darboux integrable chains

On the classification of Darboux integrable chains

Ismagil Habibullin,^a兲 Natalya Zheltukhina,^b兲and Aslı Pekcan Department of Mathematics, Faculty of Science, Bilkent University, 06800 Ankara, Turkey

共Received 25 June 2008; accepted 10 September 2008; published online 9 October 2008兲

We study a differential-difference equation of the form t_x共n+1兲= f共t共n兲,t共n+1兲, t_x共n兲兲 with unknown t=t共n,x兲 depending on x and n. The equation is called a Darboux integrable if there exist functions F共called an x-integral兲 and I 共called an n-integral兲, both of a finite number of variables x,t共n兲,t共n⫾1兲,t共n⫾2兲, ... , t_x共n兲,txx共n兲,..., such that DxF = 0 and DI = I, where D_x is the operator of total differentiation with respect to x and D is the shift operator: Dp共n兲=p共n+1兲. The Darboux integrability property is reformulated in terms of characteristic Lie alge- bras that give an effective tool for classification of integrable equations. The com- plete list of equations of the form above admitting nontrivial x-integrals is given in the case when the function f is of the special form f共x,y,z兲=z+d共x,y兲. © 2008 American Institute of Physics. 关DOI:10.1063/1.2992950兴

I. INTRODUCTION

In this paper we study integrable semidiscrete chains of the following form:

t_x共n + 1兲 = f共t共n兲,t共n + 1兲,tx共n兲兲, 共1兲 where the unknown t = t共n,x兲 is a function of two independent variables: discrete n and continuous x. Chain共1兲can also be interpreted as an infinite system of ordinary differential equations for the sequence of the variables兵t共n兲其_n=−⬁^⬁ . Here f = f共t,t1, tx兲 is assumed to be a locally analytical func- tion of three variables satisfying at least locally the condition

⳵^f

⳵^tx

⫽ 0. 共2兲

For the sake of convenience we introduce subindex denoting shifts tk= t共n+k,x兲 共keep t0= t兲 and derivatives tx=共⳵/⳵x兲t共n,x兲, txx=共⳵²/⳵^x2兲t共n,x兲, and so on. We denote through D and Dxthe shift operator and, correspondingly, the operator of total derivative with respect to x. For instance, Dh共n,x兲=h共n+1,x兲 and Dxh共n,x兲=共⳵/⳵x兲h共n,x兲. Set of all the variables 兵tk其_k=−⬁^⬁ ;兵Dx

mt其_m=1^⬁ con- stitutes the set of dynamical variables. Below we consider the dynamical variables as independent ones. Since in the literature the term “integrable” has various meanings let us specify the meaning used in the article. Introduce first notions of n- and x-integrals.¹

Functions I and F, both depending on x and a finite number of dynamical variables, are called, respectively, n- and x-integrals of共1兲 if DI = I and DxF = 0.

Definition: Chain 共1兲 is called integrable 共Darboux integrable兲 if it admits a nontrivial n-integral and a nontrivial x-integral.

Darboux integrability implies the so-called C-integrability. Knowing both integrals F and I a Cole–Hopf-type differential substitution w = F + I reduces Eq. 共1兲 to the discrete version of the D’Alembert wave equation, w_1x− wx= 0. Indeed,共D−1兲Dx共w兲=共D−1兲DxF + Dx共D−1兲I=0.

a兲Electronic mail: habibullin_i@mail.rb.ru. On leave from Ufa Institute of Mathematics, Russian Academy of Science, Chernyshevskii Str., 112, Ufa 450077, Russia.

b兲Electronic mail: natalya@fen.bilkent.edu.tr.

49, 102702-1

0022-2488/2008/49共10兲/102702/39/$23.00 © 2008 American Institute of Physics

It is remarkable that an integrable chain is reduced to a pair consisting of an ordinary differ- ential equation and an ordinary difference equation. To illustrate it, note first that any n-integral might depend only on x- and x-derivatives of the variable t, I = I共x,t,tx, txx, . . .兲, and similarly any x-integral depends only on x and the shifts, F = F共x,t,t_⫾1, t_⫾2, . . .兲. Therefore each solution of integrable chain共1兲 satisfies two equations:

I共x,t,tx,txx, . . .兲 = p共x兲, F共x,t,t_⫾1,t_⫾2, . . .兲 = q共n兲, with properly chosen functions p共x兲 and q共n兲.

Nowadays the discrete phenomena are studied intensively due to their various applications in physics. For the discussions and references we refer to the articles in Refs.1–5.

Chain 共1兲 is very close to a well studied object—the partial differential equation of the hyperbolic type

uxy= f共x,y,u,ux,uy兲. 共3兲

The definition of integrability for Eq. 共3兲 was introduced by Darboux. The famous Liouville equation uxy= e^uprovides an illustrative example of the Darboux integrable equation. An effective criterion of integrability of共3兲 was discovered by Darboux himself: Eq.共3兲 is integrable if and only if the Laplace sequence of the linearized equation terminates at both ends共see Refs.6–8兲.

This criterion of integrability was used in Ref.8where the complete list of all Darboux integrable equations of form共3兲is given.

An alternative approach to the classification problem based on the notion of the characteristic Lie algebra of hyperbolic-type systems was introduced years ago in Refs. 9 and 10. In these articles an algebraic criterion of Darboux integrability property has been formulated. An important classification result was obtained in Ref.9for the exponential system

u_xyⁱ = exp共ai1u¹+ a_i2u²+ ¯ + ainuⁿ兲, i = 1,2, ... ,n. 共4兲 It was proved that system 共4兲 is a Darboux integrable if and only if the matrix A =共aij兲 is the Cartan matrix of a semisimple Lie algebra. Properties of the characteristic Lie algebras of the hyperbolic systems

u_xyⁱ = c_jkⁱu^ju^k, i, j,k = 1,2, . . . ,n, 共5兲 have been studied in Refs.11and12. Hyperbolic systems of general form admitting integrals are studied in Ref.13. A promising idea of adopting the characteristic Lie algebras to the problem of classification of the hyperbolic systems which are integrated by means of the inverse scattering transforms method is discussed in Ref.14.

The method of characteristic Lie algebras is closely connected with the symmetry approach¹⁵ which is proved to be a very effective tool to classify integrable nonlinear equations of evolution- ary type^16–20 共see also the survey in Ref. 3 and references therein兲. However, the symmetry approach meets very serious difficulties when applied to hyperbolic-type models. After the papers in Refs.21and22it became clear that this case needs alternative methods.

In this article an algorithm of classification of integrable discrete chains of form 共1兲 is sug- gested based on the notion of the characteristic Lie algebra共see also Refs.23–25兲.

To introduce the characteristic Lie algebra L_nof 共1兲in the direction of n, note that

D^−j ⳵

⳵^t1

D^jI = 0 共6兲

for any n-integral I and jⱖ1. Indeed, the equation DI=I can be rewritten in an enlarged form as I共x,n + 1,t1, f, fx, fxx, . . .兲 = I共x,n,t,tx,txx, . . .兲. 共7兲 The left hand side DI of equality共7兲contains the variable t₁, while the right hand side does not.

Hence,共⳵/⳵^t1兲共DI兲=0, which implies

D⁻¹ ⳵

⳵^t1

DI = 0.

Proceeding this way one can easily prove共6兲from the equality D^jI = I, jⱖ1.

Define vector fields

Yj= D^−j ⳵

⳵^t1

D^j, jⱖ 1, 共8兲

and

X_j= ⳵

⳵^t−j

, jⱖ 1. 共9兲

We have YjI = 0 and XjI = 0 for any n-integral I of共1兲and jⱖ1. The following theorem 共see Ref.

24兲 defines the characteristic Lie algebra Lnof 共1兲.

Theorem 1: Equation (1) admits a nontrivial n -integral if and only if the following two conditions hold:

共1兲 Linear space spanned by the operators 兵Yj其₁^⬁is of finite dimension. We denote this dimension by N .

共2兲 Lie algebra Ln generated by the operators Y₁, Y₂, . . . , YN, X₁, X₂, . . . , XN is of finite dimen- sion. We call L_n the characteristic Lie algebra of(1) in the direction of n .

To introduce the characteristic Lie algebra Lxof共1兲in the direction of x, note that Eq.共1兲due to共2兲 can be rewritten as tx共n−1兲=g共t共n兲,t共n−1兲,tx共n兲兲. An x-integral F共x,t,t_⫾1, t_⫾2, . . .兲 solves the equation DxF = 0, i.e., K₀F = 0, where

K₀= ⳵

⳵^{x+ tx}

⳵

⳵^{t+ f}

⳵

⳵^t1

+ g ⳵

⳵^t−1

+ f₁ ⳵

⳵^t2

+ g₋₁ ⳵

⳵^t−2

+ ¯ . 共10兲

Since F does not depend on the variable txone gets XF = 0, where

X = ⳵

⳵^tx

. 共11兲

Therefore, any vector field from the Lie algebra generated by K₀and X annulates F. This algebra is called the characteristic Lie algebra L_xof chain共1兲in the x-direction.

The following result is essential. Its proof is a simple consequence of the famous Jacobi theorem共the Jacobi theorem is discussed, for instance, in Ref.10兲.

Theorem 2: Equation(1)admits a nontrivial x -integral if and only if its Lie algebra L_xis of finite dimension.

In the present paper we restrict ourselves to consideration of the existence of x-integrals for a particular kind of chain共1兲, namely, we study chains of the form

t_1x= tx+ d共t,t1兲 共12兲

admitting nontrivial x-integrals. The main result of the paper, Theorem 3 below, is the complete list of chains共12兲admitting nontrivial x-integrals.

Theorem 3: Chain (12) admits a nontrivial x -integral if and only if d共t,t1兲 is one of the following kinds:

共1兲 d共t,t₁兲=A共t−t₁兲 ,

共2兲 d共t,t1兲=c0t共t−t1兲+c2共t−t1兲²+ c₃t − c₃t₁, 共3兲 d共t,t1兲=A共t−t1兲e^␣t,

共4兲 d共t,t1兲=c4共e^␣t1− e^␣t兲+c5共e^−␣t1− e^−␣t兲 ,

where A = A共t−t1兲 is a function of␶^{= t − t}1and c₀- c₅are some constants with c₀⫽0, c4⫽0, and c₅⫽0, and␣ is a nonzero constant. Moreover, x-integrals in each of the cases are 共i兲 F = x +兰^␶du/A共u兲 if A共u兲⫽0 and F=t1− t if A共u兲⬅0 ,

共ii兲 F =共1/共−c2− c₀兲兲ln兩共−c2− c₀兲␶1/␶2+ c₂兩+1/c2ln兩c2␶1/␶^{− c}2− c₀兩 for c2共c2+ c₀兲⫽0 , F=ln␶1

− ln␶2+␶1/␶^{for c}2= 0 , and F =␶1/␶2− ln␶^{+ ln}␶1for c₂= −c₀ , 共iii兲 F =兰^␶e^−␣udu/A共u兲−兰^␶1du/A共u兲 , and

共iv兲 F =关共e^␣t− e^␣t2兲共e^␣t1− e^␣t3兲兴/关共e^␣t− e^␣t3兲共e^␣t1− e^␣t2兲兴 .

The n-integrals of chain 共12兲can be studied in a similar way by using Theorem 1, but this problem is out of the frame of the present article.

The article is organized as follows. In Sec. II, by using the properly chosen sequence of multiple commutators, a very rough classification result is obtained: function d共t,t1兲 for chain共12兲 admitting x-integrals is a quasipolynomial on t with coefficients depending on␶^{= t − t}1. Then it is observed that the exponents␣0= 0 ,␣1, . . . ,␣sin expansion共26兲cannot be arbitrary. For example, if the coefficient before e^␣0t= 1 is not identically zero, then the quasipolynomial d共t,t1兲 is really a polynomial on t with coefficients depending on ␶. In Sec. III we prove that the degree of this polynomial is at most 1. If d contains a term of the form␮共␶兲t^je^␣ktwith␣k⫽0, then j=0 共Sec. IV兲.

In Sec. V it is proved that if d contains terms with e^␣kt and e^␣jt having nonzero exponents, then

␣k= −␣j. This last case contains chains having infinite dimensional characteristic Lie algebras for which the sequence of multiple commutators grows very slowly. They are studied in Secs. VI and VII. One can find the well known semidiscrete version of the sine-Gordon 共SG兲 model among them. It is worth mentioning that in Sec. VII the characteristic Lie algebra Lxfor semidiscrete SG is completely described. The last section, Sec. VIII, contains the proof of the main theorem, Theorem 3, and here the method of constructing x-integrals is also briefly discussed.

II. A NECESSARY INTEGRABILITY CONDITION

Define a class F of locally analytical functions each of which depends only on a finite number of dynamical variables. In particular, we assume that f共t,t1, tx兲苸F. We will consider vector fields given as an infinite formal series of the form

Y =兺_−⬁^{⬁ yk}_⳵^⳵_tk^{, 共13兲}

with coefficients yk苸F. Introduce notions of linearly dependent and independent sets of vector fields共13兲. Denote through PNthe projection operator acting according to the rule

PN共Y兲 = 兺

k=−N N

yk

⳵

⳵^tk

. 共14兲

First we consider finite vector fields as

Z = 兺

k=−N N

zk

⳵

⳵^tk

. 共15兲

We say that a set of finite vector fields Z₁, Z₂, . . . , Zmis linearly dependent in some open region U if there is a set of functions ␭1,␭2, . . . ,␭mdefined on U such that the function 兩␭1兩²+兩␭2兩²+¯ +兩␭m兩²does not vanish identically and the condition

␭1Z₁+␭2Z₂+ ¯ + ␭mZm= 0 共16兲 holds for each point of region U.

We call a set of the vector fields Y₁, Y₂, . . . , Ymof form共13兲linearly dependent in region U if for each natural N the set of finite vector fields PN共Y1兲, PN共Y2兲, ... , PN共Ym兲 is linearly dependent in this region. Otherwise we call the set Y₁, Y₂, . . . , Ymlinearly independent in U.

The following proposition is very useful. Its proof is almost evident.

Proposition: If a vector field Y is expressed as a linear combination,

Y =␭1Y₁+␭2Y₂+ ¯ + ␭mY_m, 共17兲 where the set of vector fields Y₁, Y₂, . . . , Ymis linearly independent in U and the coefficients of all the vector fields Y , Y₁, Y₂, . . . , Ym belonging to F are defined in U , then the coefficients

␭1,␭2, . . . ,␭mare in F .

Below we concentrate on the class of chains of form 共12兲. For this case the Lie algebra Lx

splits down into a direct sum of two subalgebras. Indeed, since f = tx+ d and g = tx− d₋₁ one gets fk= tx+ d +兺_j=1^k dj and g_−k= tx−兺^k+1_j=1d_−k for kⱖ1, where d=d共t,t1兲 and dj= d共tj, t_j+1兲. Due to this observation the vector field K₀ can be rewritten as K₀= txX˜ +Y, with

X˜ = ⳵

⳵^t+

⳵

⳵^t1

+ ⳵

⳵^t−1

+ ⳵

⳵^t2

+ ⳵

⳵^t−2

+¯ 共18兲

and

Y = ⳵

⳵^{x+ d}

⳵

⳵^t1

− d₋₁ ⳵

⳵^t−1

+共d + d1兲 ⳵

⳵^t2

−共d−1+ d₋₂兲 ⳵

⳵^t−2

+ ¯ .

Due to the relations 关X,X˜兴=0 and 关X,Y兴=0 we have X˜=关X,K0兴苸Lx; hence Y苸Lx. Therefore Lx=兵X其丣L_x1, where L_x1is the Lie algebra generated by the operators X˜ and Y.

Lemma 1: If Eq.(12)admits a nontrivial x -integral, then it admits a nontrivial x -integral F such that⳵^F/⳵^{x = 0 .}

Proof: Assume that a nontrivial x-integral of共12兲exists. Then the Lie algebra L_x1is of finite dimension. One can choose a basis of L_x1in the form

T₁= ⳵

⳵^x+ 兺

k=−⬁

⬁

a_1,k ⳵

⳵^tk

,

Tj=_k=−⬁兺^{⬁ aj,k}_⳵^⳵_tk^{, 2ⱕ j ⱕ N.}

Thus, there exists an x-integral F depending on x , t , t₁, . . . , t_N−1, satisfying the system of equations

⳵^F

⳵^{x +}兺

k=0 N−1

a_1,k⳵^F

⳵^tk

= 0,

兺k=0 N−1

a_j,k⳵^F

⳵^tk

= 0, 2ⱕ j ⱕ N.

Due to the famous Jacobi theorem¹⁰ there is a change of variables ␪j=␪j共t,t1, . . . , t_N−1兲 that re- duces the system to the form

⳵^F

⳵^{x +}兺

k=0 N−1

˜a_1,k⳵^F

⳵␪k

= 0,

⳵^F

⳵␪k

= 0, 2ⱕ j ⱕ N − 2,

which is equivalent to

⳵^F

⳵^{x + a˜1,N−1}

⳵^F

⳵␪N−1

= 0 for F = F共x,␪N−1兲.

There are two possibilities: 共1兲 a˜1,N−1= 0 and 共2兲 a˜1,N−1⫽0. In case 共1兲, we at once have

⳵^F/⳵x = 0. In case 共2兲, F=x+H共␪N−1兲=x+H共t,t1, . . . , t_N−1兲 for some function H. Evidently, F1

= DF = x + H共t1, t₂, . . . , tN兲 is also an x-integral, and F1− F is a nontrivial x-integral not depending

on x. 䊐

Below we look for x-integrals F depending on dynamical variables t , t_⫾1, t_⫾2, . . . only共not depending on x兲. In other words, we study the Lie algebra generated by vector fields X˜ and Y˜, where

Y˜ = d ⳵

⳵^t1

− d₋₁ ⳵

⳵^t−1

+共d + d1兲 ⳵

⳵^t2

−共d−1+ d₋₂兲 ⳵

⳵^t−2

+ ¯ . 共19兲

One can prove that the linear operator Z→DZD⁻¹ defines an automorphism of the characteristic Lie algebra Lx. This automorphism plays the crucial role in all of our further considerations.

Further we refer to it as the shift automorphism. For instance, direct calculations show that

DX˜ D⁻¹= X˜ , DY˜D⁻¹= − dX˜ + Y˜ . 共20兲 Lemma 2: Suppose that a vector field of the form Z =兺a共j兲共⳵/⳵^tj兲 with the coefficients a共j兲

= a共j,t,t_⫾1, t_⫾2, . . .兲 depending on a finite number of the dynamical variables solves an equation of the form DZD⁻¹=␭Z . If for some j= j0we have a共j0兲⬅0 , then Z=0 .

Proof: By applying the shift automorphism to the vector field Z one gets DZD⁻¹

=兺D共a共j兲兲共⳵/⳵^tj+1兲. Now, to complete the proof, we compare the coefficients of ⳵/⳵^tj in the

equation兺D共a共j兲兲共⳵/⳵^tj+1兲=␭兺a共j兲共⳵/⳵^tj兲. 䊐

Construct an infinite sequence of multiple commutators of the vector fields X˜ and Y˜,

˜Y

1=关X˜,Y˜兴, Y˜k=关X˜,Y˜k−1兴 for k ⱖ 2. 共21兲 Lemma 3: We have

DY˜_kD⁻¹= − X˜^k共d兲X˜ + Y˜k, kⱖ 1. 共22兲 Proof: We prove the statement by induction on k. The base of induction holds. Indeed, by共20兲 and共21兲, we have

DY˜

1D⁻¹= D关X˜,Y˜兴D⁻¹=关DX˜D⁻¹,DY˜ D⁻¹兴 = 关X˜,− dX˜ + Y˜兴 = − X˜共d兲X˜ + Y˜1. Assuming Eq.共22兲holds for k = n − 1, we have

DY˜

nD⁻¹=关DX˜D⁻¹,DY˜

n−1D⁻¹兴 = 关X˜,− X˜ⁿ⁻¹共d兲X˜ + Y˜n−1兴 = − X˜ⁿ共d兲X˜ + Y˜n,

which finishes the proof of the lemma. 䊐

Since vector fields X, X˜ , and Y˜ are linearly independent, then the dimension of Lie algebra L_x is at least 3. By 共22兲, case Y˜

1= 0 corresponds to X˜ 共d兲=0, or dt+ dt₁= 0, which implies d = A共t

− t₁兲, where A共␶兲 is an arbitrary differentiable function of one variable.

Assume Eq.共12兲admits a nontrivial x-integral and Y˜

1⫽0. Consider the sequence of the vector fields 兵Y˜1, Y˜

2, Y˜

3, . . .其. Since Lxis of finite dimension, then there exists a natural number N such that

Y˜

N+1=␥1Y˜

1+␥2Y˜

2+ ¯ +␥NY˜

N, Nⱖ 1, 共23兲

and Y˜

1, Y˜

2, . . . , Y˜

Nare linearly independent. Therefore, DY˜

N+1D⁻¹= D共␥1兲DY˜1D⁻¹+ D共␥2兲DY˜2D⁻¹+ ¯ + D共␥N兲DY˜ND⁻¹, Nⱖ 1.

Due to Lemma 3 and共23兲the last equation can be rewritten as

− X˜^N+1共d兲X˜ +␥1Y˜

1+␥2Y˜

2+ ¯ +␥NY˜

N= D共␥1兲共− X˜共d兲X˜ + Y˜1兲 + D共␥2兲共− X˜²共d兲X˜ + Y˜2兲 + ¯ + D共␥N兲共− X˜^N共d兲X˜ + Y˜N兲.

Comparing coefficients before linearly independent vector fields X˜ ,Y˜₁, Y˜

2, . . . , Y˜

N, we obtain the following system of equations:

X˜^N+1共d兲 = D共␥1兲X˜共d兲 + D共␥2兲X˜²共d兲 + ¯ + D共␥N兲X˜^N共d兲,

␥1= D共␥1兲, ␥2= D共␥2兲, ... , ␥N= D共␥N兲.

Since the coefficients of the vector fields Y˜

jdepend only on the variables t , t_⫾1, t_⫾2, . . . the factors

␥j might depend only on these variables共see the proposition above兲. Hence the system of equa- tions implies that all coefficients ␥k, 1ⱕkⱕN, are constants, and d=d共t,t1兲 is a function that satisfies the following differential equation:

X˜^N+1共d兲 =␥1X˜ 共d兲 +␥2X˜²共d兲 + ¯ +␥NX˜^N共d兲, 共24兲 where X˜ 共d兲=dt+ dt₁. Using the substitution s = t and␶= t − t₁, Eq.共24兲can be rewritten as

⳵^N+1d

⳵^sN+1=␥1

⳵^d

⳵^s+␥2

⳵^2d

⳵^{s2+ ¯ +}␥N

⳵^Nd

⳵^{sN, 共25兲}

which implies that

d共t,t1兲 =兺_k

冉

冊

for some functions␭k,j共t−t1兲, where␣kare roots of multiplicity mkfor a characteristic equation of 共25兲.

Let␣0= 0 ,␣1, . . . ,␣sbe the distinct roots of characteristic equation共24兲. Equation共24兲can be rewritten as

⌳共X˜兲d ª X˜^m0共X˜ −␣1兲^m1共X˜ −␣2兲^m2¯ 共X˜ −␣s兲^msd = 0 共27兲 and m₀+ m₁+¯ +ms= N + 1, m₀ⱖ1.

Initiated by formula 共19兲 define a map h→Yh which assigns to any function h

= h共t,t_⫾1, t_⫾2, . . .兲 a vector field

Yh= h ⳵

⳵^t1

− h₋₁ ⳵

⳵^t−1

+共h + h1兲 ⳵

⳵^t2

−共h−1+ h₋₂兲 ⳵

⳵^t−2

+ ¯ .

For any polynomial with constant coefficients P共␭兲=c0+ c₁␭+ ¯ +cm␭^mwe have the formula

P共adX˜兲Y˜ = YP共X˜兲h, where adXY =关X,Y兴, 共28兲 which establishes an isomorphism between the linear space V of all solutions of Eq.共25兲and the linear space V˜ =span兵Y˜ ,Y˜1, . . . , Y˜

N其 of the corresponding vector fields.

Represent function 共26兲 as the sum d共t,t1兲= P共t,t1兲+Q共t,t1兲 of the polynomial part P共t,t1兲

=兺^m_j=0⁰⁻¹␭0,j共t−t1兲t^j and the “exponential” part Q共t,t1兲=兺_k=1^s 共兺^m_j=0^k−1␭k,j共t−t1兲t^j兲e^␣kt.

Lemma 4: Assume that Eq. (12)admits a nontrivial x -integral. Then one of the functions P共t,t1兲 and Q共t,t1兲 vanishes.

Proof: Assume in contrary that neither of the functions vanish. First we show that in this case algebra Lx contains vector fields T₀= YA共␶兲e^␣kt and T₁= YB共␶兲 for some functions A共␶兲 and B共␶兲.

Indeed, take T₀ª⌳0共adX˜兲Y˜ =Y_{⌳0共X˜兲d}苸Lx, where ⌳0共␭兲=⌳共␭兲/共␭−␣k兲. Evidently the function A˜ 共t,t1兲=⌳0共X˜兲d solves the equation 共X˜−␣k兲A˜共t,t1兲=⌳共X˜兲d=0 which implies immediately that A˜ 共t,t1兲=A共␶兲e^␣kt. In a similar way one shows that T₁苸Lx. Note that due to our assumption the functions A共␶兲 and B共␶兲 cannot vanish identically.

Consider an infinite sequence of the vector fields defined as follows:

T₂=关T0,T₁兴, T3=关T0,T₂兴, ... , Tn=关T0,T_n−1兴, n ⱖ 3.

One can show that

关X˜,T0兴 =␣kT₀, 关X˜,T1兴 = 0, 关X˜,Tn兴 =␣k共n − 1兲Tn, nⱖ 2,

DT₀D⁻¹= − Ae^␣ktX˜ + T₀, DT₁D⁻¹= − BX˜ + T₁,

DTnD⁻¹= Tn− 共n − 1兲共n − 2兲

2 ␣kAe^␣ktT_n−1+ bnX˜ +兺

k=0 n−2

a_k^共n兲Tk, nⱖ 2.

Since algebra Lxis of finite dimension, then there exists number N such that

T_N+1=␭X˜ +␮0T₀+␮1T₁+ ¯ +␮NTN, 共29兲 and vector fields X˜ ,T₀, T₁, . . . , TNare linearly independent. We have

DT_N+1D⁻¹= D共␭兲X˜ + D共␮0兲兵− Ae^␣ktX˜ + T₀其 + ¯ + D共␮N兲

再

2 ␣kAe^␣ktT_N−1+¯

冎

By comparing the coefficients before TN in the last equation one gets

␮N−N共N − 1兲

2 ␣kA共␶兲e^␣kt= D共␮N兲.

It follows that␮Nis a function of variable t only. Also, by applying ad_X^˜ to both sides of Eq.共29兲, one gets

N␣kT_N+1=关X˜,TN+1兴 = X˜共␭兲X˜ + 共X˜共␮0兲 +␮0␣k兲T0+ ¯ + 共X˜ 共␮N兲 +␮N共N − 1兲␣k兲TN. Again, by comparing coefficients before TN, we have

N␣k␮N= X˜ 共␮N兲 + 共N − 1兲␣k␮N, i . e . , X˜ 共␮N兲 =␣k␮N.

Therefore, ␮N= A₁e^␣kt, where A₁ is some nonzero constant, and thus A共␶兲e^␣kt= A₂e^␣kt− A₂e^␣kt1. Here A₂ is some constant. We have T₀= A₂e^␣ktX˜ −A₂S₀, where

S₀= 兺

j=−⬁

⬁

e^␣ktj ⳵

⳵^tj

.

Also,

关X˜,S0兴 =␣kS₀, DS₀D⁻¹= S₀. Consider a new sequence of vector fields,

P₁= S₀, P₂=关T1,S₀兴, P3=关T1, P₂兴, Pn=关T1, P_n−1兴, n ⱖ 3.

One can show that

关X˜,Pn兴 =␣kPn, DPnD⁻¹= Pn−␣k共n − 1兲BPn−1+ bnX˜ + a_nS₀+兺

j=2 n−2

a_j^共n兲Pj, nⱖ 2.

Since algebra L_xis of finite dimension, then there exists number M such that

P_M+1=␭^ⴱX˜ +␮2ⴱP₂+ ¯ +␮MⴱP_M, 共30兲 and fields X˜ , P₂, . . . , P_M are linearly independent. Thus,

DP_M+1D⁻¹= D共␭^ⴱ兲X˜ + D共␮2ⴱ兲兵P2+¯其 + ¯ + D共␮ⴱM兲兵PM−␣k共M − 1兲BPM−1+¯其.

We compare the coefficients before PM in the last equation and get

␮ⴱM− M␣kB共␶兲 = D共␮Mⴱ兲, 共31兲 which implies that␮Mⴱ is a function of variable t only. Also, by applying adX˜ to both sides of共30兲, one gets

␣kP_M+1=关X˜,PM+1兴 = X˜共␭^ⴱ兲X˜ + 共X˜共␮₂^ⴱ兲 +␣k␮₂^ⴱ兲P2+ ¯ + 共X˜ 共␮Mⴱ兲 +␣k␮Mⴱ兲PM.

Again, we compare the coefficients before PM and have ␣k␮Mⴱ共t兲=X˜共␮Mⴱ共t兲兲+␣k␮Mⴱ共t兲, which implies that␮Mⴱ is a constant. It follows then from共31兲that B共␶兲=0. This contradiction shows that our assumption that both functions are not identically zero was wrong. 䊐 III. MULTIPLE ZERO ROOT

In this section we assume that Eq.共12兲admits a nontrivial x-integral and that␣0= 0 is a root of the characteristic polynomial⌳共␭兲. Then, due to Lemma 4, zero is the only root and therefore

⌳共␭兲=␭^m+1. It follows from formula共26兲with m₀= m + 1 that

d共t,t1兲 = a共␶兲t^m+ b共␶兲t^m−1+ ¯ , m = m0− 1ⱖ 0.

The case m = 0 corresponds to a very simple equation, t_1x= tx+ A共t−t1兲, which is easily solved in quadratures, so we concentrate on the case mⱖ1. For this case the characteristic algebra Lx

contains a vector field T = Y_␬˜with

␬

˜ = a共␶兲t + 1 mb共␶兲.

Indeed,

T = 1 m!ad_X˜

m−1Y˜ = Y_␬˜. 共32兲

Introduce a sequence of multiple commutators defined as follows:

T₀= X˜ , T₁=关T,T0兴 = Y_{−a共␶兲}, T_k+1=关T,Tk兴, k ⱖ 0, Tk,0=关T0,Tk兴.

Note that T_1,0= 0. We will see below that the linear space spanned by this sequence is not invariant under the action of the shift automorphism Z→DZD⁻¹introduced above. We extend the sequence to provide the invariance property. We define T_␣ with the multi-index ␣. For any sequence ␣

= k , 0 , i₁, i₂, . . . , i_n−1, in, where k is any natural number, ij苸兵0;1其, denote

T_␣=

再

冎

m共␣兲 =

冦

冧

l共␣兲 = k + n + 1 − m共␣兲.

The multi-index␣is characterized by two quantities, m共␣兲 and l共␣兲, which allow to order partially the sequence兵T_␣其. We have

DT₀D⁻¹= T₀, DTD⁻¹= T −␬^{˜ T}0, DT₁D⁻¹= T₁+ aT₀. One can prove by induction on k that

DT_kD⁻¹= T_k+ aT_k−1−␬^˜_{m共␤兲=k−1}兺 ^T␤+_{m共␤兲ⱕk−2}兺 ^{␩共k,␤兲T␤. 共33兲}

In general, for any␣^,

DT_␣D⁻¹= T_␣+_{m共␤兲ⱕm共␣兲−1}兺 ^{␩共␣,␤兲T␤. 共34兲}

We can choose a system P of linearly independent vector fields in the following way:

共1兲 T and T0are linearly independent. We take them into P.

共2兲 We check whether T, T0, and T₁are linearly independent or not. If they are dependent, then P =兵T,T0其 and T1=␮T +␭T0for some functions␮and␭.

共3兲 If T, T0, and T₁are linearly independent, then we check whether T, T₀, and T₁, T₂are linearly independent or not. If they are dependent, then P =兵T,T0, T₁其.

共4兲 If T, T0, T₁, and T₂ are linearly independent, we add vector fields T_␤, m共␤兲=2, ␤苸I2

共actually, by definition I2 is the collection of such ␤兲, in such a way that J2

ª兵T,T0, T₁, T₂,艛_␤苸I2T_␤其 is a system of linearly independent vector fields and for any T_␥ with m共␥兲ⱕ2 we have T_␥=兺T_␤苸J₂␮共␥,␤兲T_␤.

共5兲 We check whether T3艛J2is a linearly independent system. If it is not, then P consists of all elements from J₂, and T₃=兺T_␤苸J₂␮共␥^,␤兲T_␤. If it is, then to the system T₃艛J2we add vector fields T_␤, m共␤兲=3,␤苸I3, in such a way that J₃ª兵T3, J₂,艛_␤苸I3T_␤其 is a system of linearly independent vector fields and for any T_␥with m共␥兲ⱕ3 we have T_␥=兺T_␤苸J3␮共␥^,␤兲T_␤. We continue the construction of system P. Since Lxis of finite dimension, then there exists such a natural number N such that we have the following:

共i兲 Tk苸 P, kⱕN.

共ii兲 m共␤兲ⱕN for any T_␤苸 P.

共iii兲 For any T_␥with m共␥兲ⱕN we have T_␥=兺T_␤苸P,m共␤兲ⱕm共␥兲␮共␥^,␤兲T_␤ and also T_N+1=␮共N+1,N兲TN+兺T_␤苸P,m共␤兲ⱕN␮共N+1,␤兲T_␤.

We then have the following:

共iv兲 For any vector field T_␣ with m共␣兲=N that does not belong to P, the coefficient ␮共␣, N兲 before T_Nin the expansion

T_␣=␮共␣^,N兲TN+_T兺

␤苸P␮共␣^,␤兲T_␤ 共35兲

is constant. Indeed, by共34兲,

DT_␣D⁻¹= T_␣+_{m共␤兲ⱕN−1}兺 ^{␩共␣,␤兲T␤=␮共␣,N兲TN+}_T兺

␤苸P␮共␣^,␤兲T_␤+_{m共␤兲ⱕN−1}兺 ^{␩共␣,␤兲T␤.}

From共35兲we also have

DT_␣D⁻¹= D共␮共␣^,N兲兲DTND⁻¹+_T兺

␤苸P

D共␮共␣^,␤兲兲DT_␤D⁻¹= D共␮共␣^,N兲兲兵TN+¯其

+_T兺

␤苸PD共␮共␣,␤兲兲兵T_␤+¯其.

By comparing the coefficients before T_N in these two expressions for DT_␣D⁻¹, we have

␮共␣,N兲 = D共␮共␣,N兲兲, which implies that␮共␣, N兲 is a constant indeed.

Lemma 5: We have a共␶兲=c0␶^{+ c}1, where c₀ and c₁ are some constants.

Proof: Since

T_N+1=␮共N + 1,N兲TN+_T兺

␤苸P␮共N + 1,␤兲T_␤, then

DT_N+1D⁻¹= D共␮共N + 1,N兲兲兵TN+¯其 +_T兺

␤苸PD共␮共N + 1,␤兲兲兵T_␤+¯其.

On the other hand,

DT_N+1D⁻¹= T_N+1+ aT_N−^˜␬_m共␤兲=N兺 ^T␤+_{m共␤兲ⱕN−1}兺 ^{␩共N + 1,␤兲T␤.}

We compare the coefficients before T_Nin the last two expressions. For Nⱖ0 the equation is

␮共N + 1,N兲 + a −^˜␬_T 兺

␤苸P,m共␤兲=N␮共␤^,N兲 = D共␮共N + 1,N兲兲. 共36兲 Denote by c = −兺T_␤苸P,m共␤兲=N␮共␤, N兲 and by ␮N=␮共N+1,N兲. By property 共iv兲, c is a constant. It follows from共36兲that␮N is a function of variables t and n only. Therefore,

a共␶兲 + c

冉

冊

By differentiating both sides of the equation with respect to t and then t₁, we have

− a⬙^共␶兲 − c

冉

冊

which implies that a⬙^共␶兲=0, or the same, a共␶兲=c₀␶^{+ c}1 for some constants c₀ and c₁. 䊐 Vector fields T₁ and T in new variables are rewritten as

T₁= 兺

j=−⬁

⬁

a共␶j兲 ⳵

⳵␶j

, 共37兲

T = −_j=−⬁兺^⬁

再

冎

再

冎

= − tT₁− 兺

j=−⬁

⬁

再

冎

where

␳j=

冦

冧

The following two lemmas are to be useful.

Lemma 6: If the Lie algebra generated by the vector fields S₀=兺^⬁_j=−⬁⳵/⳵^wj and P

=兺^⬁_j=−⬁c共wj兲共⳵/⳵^wj兲 is of finite dimension, then c共w兲 is one of the following forms:

共1兲 c共w兲=c2+ c₃e^␭w+ c₄e^−␭w,␭⫽0 , and

共2兲 c共w兲=c2+ c₃w + c₄w², where c₂− c₄are some constants.

Proof: Introduce vector fields

S₁=关S0, P兴, S2=关S0,S₁兴, ... , Sn=关S0,S_n−1兴, n ⱖ 3.

Clearly, we have

S_n= 兺

j=−⬁

⬁

c^共n兲共wj兲 ⳵

⳵^wj

, nⱖ 1. 共39兲

Since all vector fields Snare elements of Lxand Lxis of finite dimension, then there exists a natural number N such that

S_N+1=␮NSN+␮N−1S_N−1+ ¯ +␮1S₁+␮0P +␮S₀, 共40兲 and S₀, P , S₁, . . . , S_Nare linearly independent.共Note that we may assume that S0and P are linearly independent兲. Since DS0D⁻¹= S₀, DPD⁻¹= P, and DSnD⁻¹= Snfor any nⱖ1, then it follows from 共40兲that

S_N+1= D共␮N兲SN+ D共␮N−1兲SN−1+ ¯ + D共␮1兲S1+ D共␮0兲P + D共␮兲S0

and together with共40兲, it implies that␮,␮0,␮1, . . . ,␮Nare all constants.

By comparing the coefficients before⳵/⳵^{w in}共40兲one gets, with the help of共39兲, the follow- ing equality:

c^共N+1兲共w兲 =␮Nc^共N兲共w兲 + ¯ +␮1c⬘共w兲 +␮0c共w兲 +␮^.

Thus, c共w兲 is a solution of the nonhomogeneous linear differential equation with constant coeffi- cient whose characteristic polynomial is

⌳共␭兲 = ␭^N+1−␮N␭^N− ¯ −␮1␭ −␮0.

Denote by␤1,␤2, . . . ,␤tthe characteristic roots and by m₁, m₂, . . . , m_ttheir multiplicities. Follow- ing are the possibilities: