Characteristic Lie Algebras of Integrable Diﬀerential ...

arX

iv:2

102.

0735

2v1

[nl

in.S

I] 1

5 Fe

b 20

21

Characteristic Lie Algebras

of Integrable Differential-Difference Equations in 3D

I. Habibullin and A. Khakimova

Abstract

The purpose of this article is to develop an algebraic approach to the problem of integrable classification ofdifferential-difference equations with one continuous and two discrete variables. As a classification criterion,we put forward the following hypothesis. Any integrable equation of the type under consideration admitsan infinite sequence of finite-field Darboux-integrable reductions. The property of Darboux integrability of afinite-field system is formalized as finite-dimensionality condition of its characteristic Lie-Rinehart algebras.That allows one to derive effective integrability conditions in the form of differential equations on the righthand side of the equation under study. To test the hypothesis, we use known integrable equations from theclass under consideration. In this article, we show that all known examples do have this property.

1 Introduction

The theory of integrability is an important component of modern mathematical physics. At present, theproblem of classifying integrable equations with three and more independent variables is being activelystudied. For these classes of equations, in contrast to the case of two variables, there are several approachesto the problem of integrable classification (see [1–5, 7–9, 11–17, 19, 20, 31, 32, 35–38, 44, 45]) mostly based ongeometric ideas and constructions. Note that the well-known generalized symmetry method is not efficientin higher dimensions due to the non-locality problem.

The hydrodynamic reduction method is one of the most widespread among specialists in mathematicalphysics. Let us briefly explain the essence of this method for the classification of three-dimensional integrableequations, based on the following two-stage procedure. The first consists in classification of dispersionless in-tegrable systems by using the method of hydrodynamic reductions (initiated by Gibbons, Kodama and Tsarev(see [19, 20]) and developed into an efficient integrability test by the group of Ferapontov, Khusnutdinova,Novikov, Pavlov, Odesskii and Sokolov ( [12, 35–37]). Second is reconstruction of dispersive deformations ofdispersionless integrable systems based on the method of dispersive deformations of hydrodynamic reductionsdeveloped by Ferapontov, Moro and Novikov ( [14, 15]).

The article discusses a problem of developing an algebraic integrability criterion based on the Darbouxintegrable reductions for the differential-difference equations of the form:

ujn+1,x = F (ujn,x, uj+1n , ujn+1, u

jn, u

j−1n+1), −∞ < n, j <∞, (1)

where the sought function u depends on three variables, on a real x and two integers j and n. We assumethat

∂F

∂ujn,x6= 0,

∂F

∂uj+1n

6= 0,∂F

∂uj−1n+1

6= 0.

The lattice is defined on a quadrilateral graph. To preserve the parity of the forward and backward directionsin n we assume that equation (1) can be uniquely rewritten as

ujn−1,x = G(ujn,x, uj+1n−1, u

jn, u

jn−1, u

j−1n ).

Integrable equations of the form (1) have been studied by many authors (see [17] and the referencestherein). Note that, in some particular cases, equations of the form (1) can be obtained as the Backlundtransformation for the two-dimensional Toda-type chains

un,xy = f(un+1, un, un−1, un,x, un,y), −∞ < n <∞. (2)

Earlier the lattices (1) have been investigated by means of the over mentioned method of hydrodynamicreductions [17]. Within the framework of this approach, a certain class of lattices of the form (1) is studiedand integrable cases are selected.

1

http://arxiv.org/abs/2102.07352v1

In the studies [23, 24, 26, 39] a classification algorithm was developed for integrable equations with threeindependent variables based on the idea of the Darboux integrable reductions and characteristic Lie algebras.Efficiency of the method was illustrated by applying to the equations from the class (2) (see [23]).

It was observed in the article [10] that it is very convenient to study the classification problem for (2) bycombining different approaches. At the first step, equations are selected that satisfy the requirement thatthe dispersionless limit of the equation is integrable, that is its characteristic variety defines a conformalstructure, which is Einstein-Weyl, on every solution. As a result one obtaines a class of equations havingthe freedom in only one function of one variable and a few constant parameters. And then, to the obtainedequations from this class the test of Darboux integrable reductions is applied.

In this paper, we show at the level of examples that integrable differential-difference lattices of the form(1) admit Darboux integrable reductions. More precisely, the purpose of the article is to approve the following

Conjecture 1.1 A lattice of the form (1) is integrable if and only if there exists a pair of functions H(1) andH(2) of four variables such that for any choice of the integer N , a system of the hyperbolic type differential-difference equations

u1n+1,x = H(1)(u1n,x, u2n, u

1n+1, u

1n),

ujn+1,x = F j(ujn,x, uj+1n , ujn+1, u

jn, u

j−1n+1), 1 < j < N, (3)

uNn+1,x = H(2)(uNn,x, uNn+1, u

Nn , u

N−1n+1 ), −∞ < n <∞

obtained from (1) is integrable in the sense of Darboux.

Let us comment on the essence of the Conjecture 1.1. Finite-field reductions of integrable lattices thatinherit integrability are quite common, for example, they are used to construct particular solutions of chains.Such reductions are provided by imposing boundary conditions (cutoff conditions) compatible with the in-tegrability property. The most popular of this kind conditions is the periodicity closure constraint. Usually,the finite-field reductions obtained as a result of integrable cutoffs are soliton systems, and only in the excep-tional case, which is obtained by imposing degenerate cutoff conditions, do we arrive at systems integrable inthe sense of Darboux. Our hypothesis is that, on the one hand, any integrable chain of the form (1) admitsdegenerate boundary conditions, which reduce the chain to a Darboux integrable system, and on the otherhand, a chain admitting degenerate boundary conditions with this property, certainly, is integrable. Exis-tence of Darboux integrable reductions allows one to apply the technique of the characteristic Lie-Rinehartalgebras for the classifying the lattices (1).

The systematic study of partial differential equations of hyperbolic type of the form

ux,y = f(x, y, u, ux, uy),

the integration problem of which is reduced to solving ordinary differential equations began apparently inthe second half of the nineteenth century. The problem of a complete description of this class of equations,called Darboux integrable equations or equations of Liouville type was studied in the well-known works ofE. Goursat. The task of complete classification turned out to be very difficult, and later many researchersworked in this field. The reader can find an overview of the results and a description of the current state ofthe problem in a detailed article [47].

Systems of nonlinear hyperbolic equations remain less studied from the point of view of Darboux inte-grability. The exception is a class of the so-called exponential type systems [18, 21, 33, 34, 41]

uix,y = exp

∑

j

ai,juj

.

It is known that this system is Darboux integrable if and only if A = {ai,j} is the Cartan matrix of a semi-simple Lie algebra. It is remarkable that general solution for Darboux integrable systems can be constructedin a closed form (see [6, 18, 33, 43, 47]).

Characteristic algebras for the Darboux integrable scalar differential-difference equations of the form

un+1,x = un,x + f(un, un+1)

are introduced and investigated in [25, 27, 28]. In [28] the complete list of such equations is given. Darbouxintegrable systems of differential-difference equations of the exponential type are studied in [30] and [42].

We briefly discuss the content of the article. In §2 we consider finite systems of the differential-differenceequations of the form (4). We recall the definition of the x- and n-integrals and define the integrability in thesense of Darboux. We introduce the characteristic vector fields and the characteristic Lie-Rinehart algebras

2

in the directions of n and x. In §3, we study the structure of characteristic algebras and describe the action oftheir automorphisms, which will be needed when solving the classification problem. We then in §4 illustratehow to use characteristic algebra to study the problem of integrable classification of the systems of the form(3). As an example, we derive one of the necessary conditions of integrability for the system (3). First,we derive a system of nonlinear functional equations from the requirement that the characteristic algebra isfinite-dimensional, and then we reduce these functional equations to a system of differential equations. InSection 5, we study all seven known differential-difference integrable lattices with one continuous and twodiscrete independent variables. In each of these equations, we make a change of variables in order to obtainan equation that admits a degenerate cutoff. By imposing the cutoff conditions at two points, we reducethe equation to a finite-field system of differential-difference equations. Further, we show that the resultingfinite-field systems are Darboux integrable by presenting for these systems complete sets of integrals in bothcharacteristic directions (among them there are scalar equations, they are presented separately in section5.8). This circumstance confirms the correctness of the approach to the classification problem formulated inthe Conjecture 1.1 above. We use two different methods to construct integrals. In some cases, when the Laxpair satisfies the conditions of Lemma 5.1 or Lemma 5.2, it is convenient to use the Lax pair. In other cases,we use the method of characteristic Lie-Reinhart algebras. This method is always applicable for constructingintegrals, although it is more labor-consuming.

2 Darboux integrable systems of differential-difference equations

and characteristic Lie-Rinehart algebras

In this section we concentrate on a system of differential-difference equations of general form

ujn+1,x = F j(x, n, ujn,x, un, un+1), j = 1, 2, . . . , N, (4)

where un = (u1n, u2n, . . . , u

Nn ). We request that functions F j are analytic in a domain in C

2N+1, where C isthe complex plane.

We assume that system (4) can be rewritten in the converse way

ujn−1,x = Gj(x, n, ujn,x, un, un−1), j = 1, 2, . . . , N.

Variables ujn, ujn±1, u

jn±2, . . . and ujn,x, u

jn,xx, u

jn,xxx, . . . are called dynamical variables, we treat them as in-

dependent ones.In what follows we will be interested in the cases when system (4) admits nontrivial x- and n-integrals.

Let us give the necessary definitions. We call a function I = I(x, n, u, un,x, un,xx, . . .) n-integral of (4) if itsatisfies the condition DnI = I by means of the system, where Dn is the shift operator acting due to therule Dny(n) = y(n + 1). Similarly function J = J(x, n, un, un±1, un±2, . . .) is an x-integral if the equationDxJ = 0 holds, where Dx stands for the operator of the total derivative with respect to x. Integrals I = I(x)and J = J(n) depending only on x and correspondingly only on n are called trivial integrals. System(4) is called integrable in sense of Darboux if it possesses the complete set of nontrivial integrals in bothcharacteristic directions x and n. Completeness means that the number of functionally independent integralsin each direction coincides with the order N of the system.

In order to find the integrals one can use the so-called characteristic operators. Let us discuss howthese operators are derived. Assume that J is an x-integral, then according to the definition we haveDxJ(x, n, un, un±1, un±2, . . .) = 0, we use the chain rule and get

K0J =

∂

∂x+

p∑

j=1

ujn,x∂

∂ujn+ ujn+1,x

∂

∂ujn+1

+ ujn−1,x

∂

∂ujn−1

+ . . .

J = 0. (5)

We specify the operator K0 by means of the system (4) and obtain

K0 =∂

∂x+

p∑

j=1

ujn,x∂

∂ujn+ F j

n

∂

∂ujn+1

+Gjn

∂

∂ujn−1

+ F jn+1

∂

∂ujn+2

+Gjn−1

∂

∂ujn−2

+ . . . , (6)

where F jn+i = F j(x, n + i, ujn+i,x, un+i, un+i+1) and Gj

n−i = Gj(x, n − i, ujn−i,x, un−i, un−i−1). We call K0

the characteristic operator in the x direction.An important feature of the equation (5) is that the solution J does not depend on the variables

u1n,x, u2n,x, . . . , u

Nn,x despite the fact that the coefficients of the equation depend on them. Therefore in

addition to (5) any x-integral solves also a set of the following equations

XkJ = 0, k = 1, 2, . . . , N, (7)

3

where Xk = ∂∂uk

n,x. Evidently x-integral is nullified by the multiple commutators of the operators

X1, X2, . . . , XN ,K0 (8)

as well as linear combinations of the obtained operators with variable coefficients, depending on the dynamicalvariables. In other words, any x-integral belongs to the kernel of the operators in the Lie-Rinehart algebraLx, generated by the operators (8) over the ring A of locally analytic functions depending on the dynamicalvariables. We call it characteristic algebra in the direction of x. Actually algebra Lx is a finitely generatedmodule over the ring A. Here the elements in Lx can be multiplied by functions from A. The consistencyconditions:

1) [W1, aW2] =W1(a)W2 + a[W1,W2],

2) (aW1)b = aW1(b)

are assumed to be valid for any W1,W2 ∈ Lx and a, b ∈ A. In other words we request that, if W1 ∈ Lx anda ∈ A then aW1 ∈ Lx (see [40]).

The algebra Lx is of a finite dimension if it admits a basis consisting of a finite number of the elementsW1,W2, . . . ,Wk ∈ Lx such that an arbitrary operator W ∈ Lx is represented as a linear combination of theform

W = a1W1 + a2W2 + · · ·+ akWk, (9)

where the coefficients are functions a1, a2, . . . , ak ∈ A. If in (9) W = 0 then we have a1 = 0, a2 = 0, . . .,ak = 0.

Let us now discuss the n-integrals. Assume that a function I = I(x, n, un, un,x, un,xx, . . .) is an n-integralfor the system (4), i.e. the following relation holds DnI = I, or the same

I(x, n+ 1, un+1, un+1,x, un+1,xx, ...) = I(x, n, un, un,x, un,xx, . . .).

Due to the equation (4), that can be written in a shortened way as un+1,x = F (un,x, un, un+1) whereF = (F 1, F 2, . . . , FN ) we rewrite the latter as

I(x, n+ 1, un+1, F,DxF,D2xF, . . .) = I(x, n, un, un,x, un,xx, . . .). (10)

The problem of finding n-integrals is much more difficult than finding x-integrals, indeed, in this case, oneneeds to solve not a differential but a functional equation (10). Below we introduce characteristic operatorsthat allow to study completely this equation. It is easily observed that the right hand side of (10) does notcontain dependence on the variable un+1 = (u1n+1, u

2n+1, . . . , u

Nn+1) therefore the left hand side satisfies the

relations ∂

∂uj

n+1

DnI = 0, which evidently implies

Yj,1I := D−1n

∂

∂ujn+1

DnI = 0 for j = 1, 2, . . . , N. (11)

It can be proved by a direct computation that characteristic operators defined in (11) act on the dynamicalvariables un, un,x, un,xx, . . . as vector fields of the form

Yj,1 =∂

∂ujn+

N∑

i=1

D−1n

(

∂F in

∂ujn+1

)

∂

∂uin,x+D−1

n

(

∂F in,x

∂ujn+1

)

∂

∂uin,xx+ . . . , (12)

where F in,x := DxF

in. We set Yj,0 = ∂

∂uj

n+1

and then rewrite representation as follows

Yj,1 =∂

∂ujn+

N∑

i=1

D−1n

(

Yj,0(

F in

)) ∂

∂uin,x+D−1

n

(

Yj,0(

F in,x

)) ∂

∂uin,xx+ . . . .

In contrast to the case of the previously studied x-integrals, in this case there are additional characteristicoperators, which also annul I. For any natural number k, the following relation holds:

Yj,kI := D−kn

∂

∂ujn+1

DknI = 0 for j = 1, 2, . . . , N. (13)

We give uniform representations for all characteristic operators (13) for k ≥ 2

Yj,k =

N∑

i=1

D−1n

(

Yj,k−1

(

F in

)) ∂

∂uin,x+D−1

n

(

Yj,k−1

(

F in,x

)) ∂

∂uin,xx+ . . . . (14)

4

Since the coefficients of the equations (13) depend on the variables ujn−s for 1 ≤ s ≤ k and 1 ≤ j ≤ N , whilethe solution to the equation does not depend on them, then such equations

Xi,sI = 0,

where Xi,s =∂

∂uin−s

should hold for these values of i and s.

It is important that the operators Yj,i commute with each other.

Lemma 2.1 For any i, i′ ≥ 0 and j, j′ such that 1 ≤ j, j′ ≤ N the relation [Yj,i, Yj′,i′ ] = 0 holds.

Proof 2.1 Let us first assume that i′ = 0, then obviously

[Yj,i, Yj′,0] =

[

Yj,i,∂

∂uj′

n+1

]

= 0

since the coefficients of the vector-field Yj,i do not depend on the variables ujn+1 for any j ∈ [1, N ] due toexplicit formula (14). Now consider general case assuming that i = i′ + k, k ≥ 0:

[Yj,i′+k, Yj′,i′ ] =[

D−i′

n Yj,kDi′

n , D−i′

n Yj′,0Di′

n

]

= D−i′

n [Yj,k, Yj′,0]Di′

n .

Latter vanishes due to the previous step.

Explicit expressions for the commutators of the basic operators Yj,k and Xj,k with the operator Dx ofthe total derivative with respect to x provide an effective tool for studying the system (4). It can easily beapproved that the following operator identities hold

[Dx, Yj,0] = −

N∑

i=1

∞∑

k=1

Yj,0(

Dk−1n F i

n

) ∂

∂uin+k

, (15)

[Dx, Yj,1] = −

N∑

i=1

(

D−1n

(

Yj,0(

F in

))

Yi,1 +

∞∑

k=1

Yj,1(

Dk−1n F i

n

) ∂

∂uin+k

+

∞∑

k=1

Yj,1(

D1−kn Gi

n

) ∂

∂uin−k

)

, (16)

[Dx, Yj,m] = −

N∑

i=1

(

m∑

s=1

D−sn

(

Yj,m−s

(

F in

))

Yi,s +

∞∑

k=1

Yj,m(

Dk−1n F i

n

) ∂

∂uin+k

+

∞∑

k=1

Yj,m(

D1−kn Gi

n

) ∂

∂uin−k

)

,

(17)

[

Dx, Xj,1

]

= −N∑

i=1

∞∑

k=1

Xj,1

(

D1−kn Gi

n

) ∂

∂uin−k

. (18)

In a particular scalar case of the system (4) when N = 1 formulas (15)–(18) have been derived in [28].Let Ak be a ring of locally analytic functions of the variables un−k, un−k+1, . . . , un; unx, un,xx, un,xxx, . . ..

Here we formulate two theorems relating integrability in the sense of Darboux and properties of the charac-teristic Lie-Rinehart algebras.

Theorem 2.1 System (4) admits a complete set of n-integrals if and only if the following two conditionshold:

1) The linear space V spanned by the operators {Yi,s} has finite dimension, which we denote by N1.Assume that Z1, Z2, . . . , ZN1

constitute a basis in V , such that for any Z ∈ V we have an expansion

Z = λ1Z1 + λ2Z2 + . . .+ λN1ZN1

,

We emphasize that the coefficients in this expansion, as a rule, are not constant, they are analytic functions ofdynamic variables. Then due to construction of the space V there exists a number N2, such that [Zj, Xi,s] = 0for all j = 1, 2, . . . , N1, i = 1, 2, . . . , N and s > N2.

2) The Lie-Rinehart algebra Ln generated by the operators {Zj}N1

j=1 and {Xi,s}N2,Ns=1,i=1 over the ring AN2

has a finite dimension.

Theorem 2.2 System (4) admits a complete set of x-integrals if and only if the characteristic Lie-Rinehartalgebra Lx has a finite dimension.

The proof of these two theorems is beyond the scope of this paper. In the particular case when N = 1Theorems 2.1, 2.2 are proved in [29]. For systems of differential equations of hyperbolic type, similar statementis proved in [46]. From these theorems we derive

Corollary 2.1 System (4) is integrable in the sense of Darboux if and only if both characteristic algebrasLx and Ln have finite dimension.

5

3 Some properties of the characteristic algebras and their potentialapplications

In this section we will discuss some important properties of the characteristic algebras. Here we outline anapproach to the problem of integrable classification of the lattices from the class (1) based on algebraic ideas.Let us first concentrate on the algebra Lx for the system (4), generated as it was discussed in §2 above bythe operators X1, X2, . . . , XN ,K0. It is easy to see that the operator Dx of the total derivative with respectto x acts on the functions of the dynamical variables un,x, un, un±1, un±2, . . . as follows

DxH(un,x, un, un±1, un±2, . . .) =

N∑

j=1

ujn,xxXj +K0

H(un,x, un, un±1, un±2, . . .),

where the operators K0 and Xj have already been defined above (see (6), (7)). Since the operators Dn andDx commute with each other we have a relation DnDxD

−1n = Dx or, the same

Dn

N∑

j=1

ujn,xxXj +K0

D−1n =

N∑

j=1

ujn,xxXj +K0. (19)

We simplify the left hand side of (19) due to the relations Dnujn,xx = ujn+1,xxDn and due to

ujn+1,xx = DxFjn =

∂F jn

∂ujn,x· ujn,xx +K0(F

jn).

As a result we obtain the relation

N∑

j=1

(

∂F jn

∂ujn,x· ujn,xx +K0

(

F jn

)

)

DnXjD−1n +DnK0D

−1n =

N∑

j=1

ujn,xxXj +K0. (20)

Let us define an automorphism of the algebra Lx, acting according to the formula

Z → DnZD−1n . (21)

It is remarkable that equation (20) allows one to describe the action of the automorphism on the basicoperators. Indeed, since variables u1n,xx, u

2n,xx, . . . , uNn,xx are regarded as independent ones, we can compare

the coefficients in front of these variables in (20) and obtain the formulas

DnXjD−1n =

1∂F

jn

∂ujn,x

Xj, (22)

DnK0D−1n = K0 −

N∑

j=1

K0(Fjn)

∂Fjn

∂ujn,x

Xj . (23)

Elements of the algebra Lx are vector fields with infinite number of components. Therefore, to proverelations of the form Y = 0 for Y ∈ Lx, it is necessary to check an infinite set of conditions. The followinglemma provides a convenient tool for exploring such questions.

Lemma 3.1 Assume that the vector field

K =

N∑

j=1

∞∑

k=1

(

αj(k)∂

∂ujn+k

+ αj(−k)∂

∂ujn−k

)

,

solves the equationDnKD

−1n = hK, (24)

where the factor h is a function of the dynamical variables, then K = 0.

Proof 3.1 First, we find an explicit representation of the operator DnKD−1n and then by substituting the

result into equation (24) we obtain

Dn(αj(−1))

N∑

s=1

Dn

(

∂

∂ujn−1

Gsn

)

Xs +Dn(αj(−1))

∂

∂ujn+Dn(α

j(−2))∂

∂ujn−1

6

+

∞∑

k=2

Dn(αj(k − 1))

∂

∂ujn+k

+Dn(αj(−k − 1))

∂

∂ujn−k

(25)

= h

(

∞∑

k=1

αj(k)∂

∂ujn+k

+ αj(−k)∂

∂ujn−k

)

for any j from the segment 1 ≤ j ≤ N .By comparing the coefficients before the operators ∂

∂uj

n+k

and ∂

∂uj

n−k

in the equation (25) one can easily

prove that αj(k) = 0 for all values of k.

For the case of the algebra Ln we have a similar statement.

Lemma 3.2 Suppose that the vector field

Y =

N∑

j=1

(

αj(1)∂

∂ujn,x+ αj(2)

∂

∂ujn,xx+ αj(3)

∂

∂ujn,xxx+ · · ·

)

(26)

solves the equation[Dx, Y ] = hY, (27)

where h is a function of the dynamical variables, then Y = 0.

Proof 3.2 From (26) and (27) we easily obtain the equation

N∑

j=1

(

−αj(1)∂F j

n

∂ujn,x

∂

∂ujn+1

− αj(1)∂

∂ujn+ (Dxα

j(1)− αj(2))∂

∂ujn,x+ (Dxα

j(2)− αj(3))∂

∂ujn,xx+ · · ·

)

= h

N∑

j=1

(

αj(1)∂

∂ujn,x+ αj(2)

∂

∂ujn,xx+ αj(3)

∂

∂ujn,xxx+ · · ·

)

.

Now we compare the coefficients before the operators ∂

∂uj

n+1

, ∂

∂ujn

, ∂

∂ujn,x

, ∂

∂ujn,xx

, . . . to make sure that

αj(k) = 0 for all k. That completes the proof of Lemma 3.2.

4 Characteristic algebras for the reduced system (3)

In this section we concentrate on the differential-difference system (3) being a reduction of the differential-difference equation (1). System (3) is a particular case of the general system (4), where assumed that

F 1 = H(1)(u1n,x, u2n, u

1n+1, u

1n),

F j = F (ujn,x, uj+1n , ujn+1, u

jn, u

j−1n+1) for 2 ≤ j ≤ N − 1,

FN = H(2)(uNn,x, uNn+1, u

Nn , u

N−1n+1 ).

Suppose that system (3) is integrable, then its characteristic algebra Lx has a finite dimension. The problemis figuring out what this means in terms of the right hand side of the differential-difference equation (1),i.e. in terms of the function F (ujn,x, u

j+1n , ujn+1, u

jn, u

j−1n+1). Below we make the first step towards solving the

problem. We define a sequence of the operators in Lx according to the rule

K1 = [Xj ,K0], K2 = [Xj ,K1], . . . , Km = [Xj ,Km−1], . . . . (28)

Since the algebra Lx is finite dimensional then there exists an integer M such that operatorKM is representedas a linear combination of the preceding members of the sequence:

KM = λKM−1 + λ1KM−2 + · · ·+ λM−1K1, (29)

where the operators KM−1,KM−2, . . . ,K1 are linearly independent.Explicit formula (6) shows that operator K1 = [Xj ,K0] has the following coordinate representation

K1 =∂

∂ujn+Xj(F

jn)

∂

∂ujn+1

+Xj(Gjn)

∂

∂ujn−1

+Xj(Fjn+1)

∂

∂ujn+2

+Xj(Gjn−1)

∂

∂ujn−2

+ . . . ,

7

while for m ≥ 2 we have

Km = Xmj (F j

n)∂

∂ujn+1

+Xmj (Gj

n)∂

∂ujn−1

+Xmj (F j

n+1)∂

∂ujn+2

+Xmj (Gj

n−1)∂

∂ujn−2

+ . . . . (30)

Since operator K1 contains the term ∂

∂ujn

and the other operators in (29) do not, then the coefficient λM−1

vanishes.Automorphism (21) provides an effective tool for studying the problem of decomposition (29). Above in

(22), (23) we have already described the action of the automorphism on the basic operators. We can derivesimilar formulas for the members of the sequence (28)

DnK1D−1n = qjK1 − q2jK1(F

jn)Xj − qj−1F

j−1

n,ujn

qjXj−1 − qj+1Fj+1

n,ujn

Xj+1, (31)

DnK2D−1n = q2jK2 + qjXj(qj)K1 + r2Xj − qj−1F

j−1

ujn

qjXj(qj)Xj−1, (32)

DnK3D−1n = q3jK3 + 3q2jX(qj)K2 + (qjXj)

2(qj)K1 + r3Xj − qj−1Fj−1

n,ujn

{

(qjXj)2(qj)

}

Xj−1. (33)

Here qj =(

∂F jn

∂ujn,x

)−1

, and r2, r3 are functions on qj and its derivatives. For the general case we can prove

by induction that

DnKmD−1n =qmj Km + pmKm−1 + smKm−2 + . . .

+ (qjXj)m−1(qj)K1 + rmXj + qj−1F

j−1

n,ujn

{

(qjXj)m−1(qj)

}

Xj−1,(34)

where

pm =1

2m(m− 1)qm−1

j Xj(qj),

sm =1

6m(m− 1)(m− 2)qm−1

j X2j (qj) +

1

24m(m− 1)(m− 2)(3m− 5)qm−2(Xj(qj))

2.

Explicit expression for rm is rather complicated and we do not specify it.Let us apply automorphism (21) to both sides of the equation (30) and get

qMj KM + pMKM−1 + sMKM−2 + . . .

+ (qjXj)M−1(qj)K1 + rMXj + qj−1F

j−1

n,ujn

{

(qjXj)M−1(qj)

}

Xj−1

= Dn(λ)(

qM−1j KM−1 + . . .

)

+Dn(λ1)(

qM−2j + . . .

)

+ . . . .

(35)

We first replace KM due to (30) and then collect in (35) the coefficients in front of the linearly independentoperators KM−1, KM−2, . . . , K1:

KM−1 : qMj λ+ pM = Dn(λ)qM−1j , (36)

KM−2 : qMj λ1 + sM = Dn(λ)pM−1 +Dn(λ1)qM−2j , (37)

. . . . . . . . . . . . . . .

K1 : (qjXj)M−1qj = Dn(λ)(qjXj)

M−2qj + . . .+Dn(λ)qjXj(qj). (38)

Due to the explicit expression (30) the coefficients of the operators K2, K3, . . . , KM depend on ujn,x and

on the variables uj+1n , ujn, uj−1

n and their shifts on n. The factors λ, λ1, . . . , λM−2 might depend only onthese variables. However we have relations (36)–(38) which provide an additional restriction for the factors.Indeed the coefficients qj , pm, sm in (36)–(38) depend only on the variables ujn,x, u

j+1n , ujn+1, u

jn, uj−1

n+1, hence

λ, λ1, . . . , λM−2 might depend only on the variables ujn,x, ujn.

Relations (36)–(38) determine a system of functional equations with the set of unknown functions λ, λ1,. . . , λM−2 depending on two variables ujn,x, ujn. Indeed, each of these equations contains the unknowns taken

at different points (ujn,x, ujn) and

(

Dnujn,x, Dnu

jn

)

=(

F(

ujn,x, uj+1n , ujn+1, u

jn, u

j−1n+1

)

, ujn+1

)

. An important

peculiarity of the system (36)–(38) is that it is highly overdetermined since the coefficients of the equationsdepend on five independent variables while the solutions depend only on two of them.

Let us finalize the reasoning above as a statement.

Theorem 4.1 (Necessary condition of integrability of system (3) and hence due to Conjec-ture 1.1 of the equation (1).) If system (3) admits the complete set of x-integrals then the overdeterminedsystem of equations (36)–(38) admits a solution (λ, λ1, . . . , λM−2) depending only on ujn,x, u

jn for any j from

the segment 1 ≤ j ≤ N .

8

Below we show how to reduce the system (36)–(38) to a system of differential equations by using thecharacteristic operators in the direction of n. We start with the equation (36) that is specified to the form

λ+ ε1

qjXj(qj) = Dn(λ)

1

qj,

where ε = M(M−1)2 . Since qj =

(

∂F jn

∂ujn,x

)−1

equation reduces to

λ∂F j

n

∂ujn,x− ε

∂2F jn

∂(ujn,x)2= Dn(λ)

(

∂F jn

∂ujn,x

)2

. (39)

We can rewrite the latter equation as

λ− ε∂

∂ujn,xlog

∂F jn

∂ujn,x= Dn(λ)

∂F jn

∂ujn,x. (40)

Let us apply the operator ∂

∂uj

n+1

to both sides of the equation (40). By taking into account that λ =

λ(

ujn,x, ujn

)

we obtain

− ε∂

∂ujn+1

∂

∂ujn,x

(

log∂F j

n

∂ujn,x

)

=∂

∂ujn+1

(

Dn(λ)∂F j

n

∂ujn,x

)

. (41)

As it was discussed above we have the relation D−1n

∂

∂uj

n+1

Dn = Yj,1 (see formula (12)) when the operators

act on the variables ujn, ujn−1, ujn,x. Since ∂

∂ujn,x

= Xj we can represent (41) in the form

− εYj,1{

D−1n Xj logXj(F

jn)}

= Yj,1{

λD−1n (Xj(F

jn))}

. (42)

In what follows we need some rather simple formulas which are produced by the obvious identity

ujn,x = Gj(

F j(

ujn,x, uj+1n , ujn+1, u

jn, u

j−1n+1

)

, uj+1n , ujn+1, u

jn, u

j−1n+1

)

.

Differentiation of the identity with respect to the variable ujn,x gives rise to equation

1 =∂DnG

jn

∂F jn

×∂F j

n

∂ujn,x

which implies

1 =

(

Dn

∂Gjn

∂ujn,x

)(

∂F jn

∂ujn,x

)

or the same

D−1n XjF

jn =

1

XjGjn

. (43)

By combining (22) and (43) we find

D−1n XjDn =

1

XjGjn

Xj . (44)

By differentiating identity with respect to ujn+1 we obtain an equation

0 = Dn

(

∂Gjn

∂ujn

)

+Dn

(

∂Gjn

∂ujn,x

)

∂F jn

∂ujn+1

,

which implies

D−1n

∂F jn

∂ujn+1

= −∂Gj

n/∂ujn

∂Gjn/∂u

jn,x

.

Now equation (42) is easily converted due to (43), (44) to the form

εYj,1

(

X2jG

jn

(XjGjn)2

)

= Yj,1

(

λ

XjGjn

)

. (45)

9

Let as replace the characteristic operator Yj,1 by its explicit representation (12). Since in (45) the operator

is applied to functions depending on the variables ujn,x, uj+1n−1, u

jn, ujn−1, u

j−1n only we omit the terms which

we do not use. More precisely we use the relation that holds for the function h depending on these variables

Yj,1h =

(

∂

∂ujn−D−1

n

(

∂F jn

∂ujn+1

)

∂

∂ujn,x

)

h.

Therefore equation (45) gets the form

(

∂

∂ujn−

∂Gjn/∂u

jn

∂Gjn/∂u

jn,x

∂

∂ujn,x

)

λ

∂Gjn/∂u

jn,x

− ε∂2Gj

n/∂(ujn,x)

2

(

∂Gjn/∂u

jn,x

)2

= 0. (46)

Thus the functional equation (36) produces a differential consequence (46). In a similar way we can derivedifferential consequences for the other functional equations of the system (36)–(38). For example (37) implies

(

∂

∂ujn−

∂Gjn/∂u

jn

∂Gjn/∂u

jn,x

∂

∂ujn,x

)(

δX3

j (Gjn)

(Xj(Gjn))3

+ γ(X2

j (Gjn))

2

(Xj(Gjn))4

− ε1λX2

j (Gjn)

(Xj(Gjn))2

+λ1

(Xj(Gjn))2

)

= 0,

where ε1 = (M−1)(M−2)2 , δ = M(M−1)(M−2)

6 , γ = M(M−1)(M−2)(M−3)8 .

By using the identity (qjXj)mqj = Xm+1

j (Gjn), that is easily proved by induction one can rewrite the last

equation of the system as follows

(

XMj − λXM−1

j − λ1XM−2j − . . .− λM−2X

2j

)

Gjn = 0.

Finally we can state that the overdetermined system of functional equations (36)–(38) is reduced to anoverdetermined system of differential equations, that is certainly more simple. An equation (46) is of a specialinterest, it should admit a solution λ = λ

(

ujn, ujn,x

)

depending only on two variables ujn and ujn,x. It is a

severe requirement since the coefficients of the equation (46) depend generally on five variables ujn,x, uj+1n−1,

ujn, ujn−1, uj−1n .

Example 4.1 We consider an illustrative example by taking a scalar equation of the form

un+1,x = un,xu2n+1. (47)

Here Fn = un,xu2n+1,

∂Fn

∂un,x= u2n+1 and ∂2Fn

∂(un,x)2= 0. Equation (39) turns into

Dn(λ)

λ=

1

u2n+1

(48)

Due to equation (47) we can rewrite (48) as

Dn(λ)

λ=

un,xun+1,x

.

Evidently it has a solution of the necessary form λ = cun,x

where c is a constant.

5 Examples, approving the Conjecture 1.1

The aim of this section is discussing the integrable lattices of the form (1) given in [17]. Since these equationsare written as

vsk+1,x = F (vs+1k,x , v

sk, v

sk+1, v

s+1k , vs+1

k+1), −∞ < s, k <∞,

at first we convert them into the form (1) by a linear transformation of the independent variables: vsk = uj+1n ,

n = k, j = −k − s. Below we give the list of integrable differential-difference equations from [17] rewrittenin the new variables

1) ujn+1,x = ujn,x + euj

n+1−uj+1

n − euj−1

n+1−uj

n ; (49)

2) ujn+1,x = ujn,x +eu

j−1

n+1

euj

n+1

−eu

j−1

n+1

eujn

−eu

jn

euj+1n

+eu

j

n+1

euj+1n

; (50)

10

3) ujn+1,x = ujn,x

(

ujn+1

)2

uj−1n+1u

j+1n

; (51)

4) ujn+1,x = ujn,x

(

ujn+1 − uj+1n

)

(

uj−1n+1 − ujn

) ; (52)

5) ujn+1,x = ujn,x

ujn+1

(

ujn+1 − uj+1n

)

uj+1n

(

uj−1n+1 − ujn

) ; (53)

6) ujn+1,x = ujn,x

(

uj−1n+1 − ujn+1

)(

ujn+1 − uj+1n

)

(

uj−1n+1 − ujn

)(

ujn − uj+1n

) ; (54)

7) ujn+1,x = ujn,x

sinh(

uj−1n+1 − ujn+1

)

sinh(

ujn+1 − uj+1n

)

sinh(

uj−1n+1 − ujn

)

sinh(

ujn − uj+1n

) (55)

their Lax pairs, also found in [17] are respectively, of the form

1)

{

ϕjn+1 = −ϕj+1

n + eujn−u

j

n+1ϕjn,

ϕjn,x = −eu

j−1n −uj

nϕj−1n ;

(56)

2)

{

ϕjn+1 = −e−u

j

n+1+uj

n

(

ϕj+1n − ϕj

n

)

+ ϕj+1n ,

ϕjn,x = eu

j−1n −uj

n

(

ϕj−1n − ϕj

n

)

;(57)

3)

ϕjn+1 = −

uj

n+1

uj+1n

(

ϕjn − ϕj+1

n

)

,

ϕjn,x = −

ujn,x

ujn

(

ϕj−1n − ϕj

n

)

;(58)

4)

{

ϕjn+1 =

(

uj+1n − ujn+1

)

ϕjn + ϕj+1

n ,

ϕjn,x = −ujn,xϕ

j−1n ;

(59)

5)

ϕjn+1 =

(

1−uj

n+1

uj+1n

)

ϕjn −

uj

n+1

uj+1n

ϕj+1n ,

ϕjn,x =

ujn,x

ujn

(

ϕj−1n + ϕj

n

)

;

(60)

6)

ϕjn+1 =

uj

n+1−uj+1

n

ujn−u

j+1n

ϕjn +

(

1−uj

n+1−uj+1

n

ujn−u

j+1n

)

ϕj+1n ,

ϕjn,x =

ujn,x

uj−1n −u

jn

(

ϕj−1n + ϕj

n

)

;

(61)

7)

ϕjn+1 = e

2(ujn+1

−uj+1n)−1

e2(uj

n−uj+1n )

−1

ϕjn +

(

1− e2(uj

n+1−u

j+1n )

−1

e2(uj

n−uj+1n )

−1

)

ϕj+1n ,

ϕjn,x =

2ujn,x

e2(uj−1

n −ujn)

−1

(

ϕj−1n − ϕj

n

)

.(62)

It can be shown that equations (49)-(55) admit infinite sequences of suitable finite field reductions beingintegrable in the sense of Darboux systems of differential-difference hyperbolic type equations. Simultane-ously, the Lax pairs (56)-(62) of equations also pass into the Lax pairs of the corresponding reductions. Thecrucial point of our algorithm is finding a special constraint consistent with the lattice that divides the latticeinto two independent parts. We call such kind of constraint a degenerate boundary condition. Below we lookfor the degenerate boundary conditions for the equations (49)-(55) and appropriate boundary conditions fortheir Lax pairs.

The Lax pair is an important attribute of the integrable soliton equations. They also have useful appli-cations for Darboux integrable systems. For example, in some cases, Lax pairs provide an effective tool forconstructing characteristic integrals. We will now outline a simple algorithm suitable for this purpose.

Assume that system of equations (4) admits a Lax pair, i.e. it is a compatibility condition of a pair ofsystems of linear equations

Φn+1 = UnΦn, (63)

Φn,x = VnΦn, (64)

11

where the potentials Un and Vn of the linear systems are matrices with the following triangular structure

Un =

a11,n a12,n . . . a1N,n

0 a22,n . . . a2N,n

. . . . . . . . . . . .0 0 . . . aNN,n

, (65)

Vn =

b11,n 0 . . . 0b21,n b22,n . . . 0. . . . . . . . . . . .bN1,n bN2,n . . . bNN,n

. (66)

Let P(k)n be a product of the matrices

P (k)n = Un+kUn+k−1 · · ·Un,

then obviously we haveDxP

(k)n = Vn+k+1P

(k)n − P (k)

n Vn. (67)

Evidently P(k)n is an upper triangular matrix

P (k)n =

p(k)11,n p

(k)12,n . . . p

(k)1N,n

0 p(k)22,n . . . p

(k)2N,n

. . . . . . . . . . . .

0 0 . . . p(k)NN,n

.

In the equality (67), we select the matrix elements located at the intersection of the first row and the lastcolumn. As a result, we obtain a scalar equality of the form

Dxp(k)1N,n = p

(k)1N,n(D

k+1n b11 − bNN ). (68)

Now we can conclude:

Lemma 5.1 If the entries of the matrix (66) satisfy the relations Dnb11 = b11 and b11 = bNN then function

J = p(k)1N,n is an x-integral.

Proof 5.1 In such a case (68) implies Dxp(k)1N,n = 0.

For constructing n-integrals we use the higher order derivatives of the equation system (64) with respectto x. Evidently they solve linear systems as

Φn,xx = (Vn,x + V 2n )Φn,

Φn,xxx = (Vn,xx + 2VnVn,x + VnVn,x + V 3n ))Φn.

For arbitrary k we find

Dkx(Φn) = R(k)

n

(

Vn, Vn,x, . . . , D(k−1)x (Vn)

)

Φn,

where R(k)n is a polynomial with constant coefficients on all of its arguments

R(k)n =

r(k)11,n 0 . . . 0

r(k)21,n r

(k)22,n . . . 0

. . . . . . . . . . . .

r(k)N1,n r

(k)N2,n . . . r

(k)NN,n

.

By applying the shift operator Dn to the latter equation we obtain

DkxΦn+1 = Dn(R

(k)n )UnΦn. (69)

On the other hand by differentiating (63) with respect to x k times we find

DkxΦn+1 =

k∑

j=0

cjkDjx(Un)(R

(k−j)n )Φn. (70)

12

Equations (69) and (70) evidently imply

Dn(R(k)n )Un =

k∑

j=0

cjkDjx(Un)(R

(k−j)n ). (71)

Let us pass in the equality (71), to the matrix elements located at the left lower corner. As a result we arriveat the equation

Dn(r(k)N1,n)a11 = aNNr

(k)N1,n + aNN,xr

(k−1)N1,n + aNN,xxr

(k−2)N1,n + · · ·+D(k)

x (aNN ). (72)

Lemma 5.2 If the entries of the matrix (65) satisfy the conditions Dxa11 = 0 and a11 = aNN then function

I = r(k)N1,n is an n-integral.

Proof 5.2 Proof follows right away from equation (72).

5.1 Reductions of the equation (49)

To find the desired degenerate boundary condition for (49) we make the following change of the variables:

ujn = vjn − log ε for j > 0, ε > 0,

u0n = v0n,

ujn = vjn + log ε for j < 0.

Then (49) converts into

v0n+1,x = v0n,x + εev0n+1−v1

v − εev−1

n+1−v0

n ,

v1n+1,x = v1n,x + ev1n+1−v2

n − εev0n+1−v1

n ,

v−1n+1,x = v−1

n,x + εev−1

n+1−v0

n − ev−2

n+1−v−1

n ,

vjn+1,x = vjn,x + evj

n+1−vj+1

n − evj−1

n+1−vj

n , for |j| > 1.

Now we take the limit for ε → 0 and obtain an equation v0n+1,x = v0n,x and two semi-infinite lattices whichare not related to each other:

v1n+1,x = v1n,x + ev1n+1−v2

n ,


n+1−vj+1

n − evj−1

n+1−vj

n , for j ≥ 2,

and

v−1n+1,x = v−1

n,x − ev−2

n+1−v−1

n ,


n+1−vj+1

n − evj−1

n+1−vj

n , for j ≤ −2.

By applying this manipulation to the lattice at two fixed points j = −1 and j = N + 1 we obtain a finitefield system of the form (3) which is a desired reduction of the equation (49):

u0n+1,x = u0n,x + eu0n+1−u1

n ,

ujn+1,x = ujn,x + euj

n+1−uj+1

n − euj−1

n+1−uj

n , 1 < j < N − 1,

uNn+1,x = uNn,x − euN−1

n+1−uN

n .

(73)

In [42] it was proved that system (73) is integrable in the sense of Darboux for arbitrary natural N . Werewrite the Lax pair (56) by means of the obtained boundary conditions

ϕ0n+1 = −ϕ1

n + eu0n−u0

n+1ϕ0n,

ϕjn+1 = −ϕj+1

n + eujn−u

j

n+1ϕjn, 1 < j < N − 1,

ϕNn+1 = eu

Nn −uN

n+1ϕNn ,

ϕ0n,x = 0,

ϕjn,x = −eu

j−1n −uj

nϕj−1n , 1 < j < N − 1,

ϕNn,x = −eu

N−1n −uN

n ϕN−1n .

When deriving it from (56) we set ϕ−1n = 0, ϕN+1

n = 0. Let us study in more details the case N = 2 thatcorresponds to the system

u0n+1,x = u0n,x + eu0n+1−u1

n ,

u1n+1,x = u1n,x + eu1n+1−u2

n − eu0n+1−u1

n ,

u2n+1,x = u2n,x − eu1n+1−u2

n

(74)

13

admitting the Lax pair

ϕ0n+1 = −ϕ1

n + eu0n−u0

n+1ϕ0n,

ϕ1n+1 = −ϕ2

n + eu1n−u1

n+1ϕ1n,

ϕ2n+1 = eu

2n−u2

n+1ϕ2n,

ϕ0n,x = 0,

ϕ1n,x = −eu

0n−u1

nϕ0n,

ϕ2n,x = −eu

1n−u2

nϕ1n.

(75)

We give here x-integrals and n-integrals of system (74) found earlier in [42]

J1 = eu0n+2−u0

n+3 + eu1n+1−u1

n+2 + eu2n−u2

n+1 ;

J2 = eu0n+1−u0

n+2+u1n+1−u1

n+2 + eu0n+1−u0

n+2+u2n−u2

n+1 + eu1n+1−u1

n+2+u2n−u2

n+1 ;

J3 = eu0n−u0

n+1+u1n−u1

n+1+u2n−u2

n+1

and, respectively

I1 = u0n,x + u1n,x + u2n,x;

I2 = u1n,xx + 2u0n,xx + u0n,xu1n,x + u0n,xu

2n,x + u1n,xu

2n,x;

I3 = u0n,xxx + u0n,xu1n,xx + u0n,xxu

1n,x + u0n,xxu

2n,x + u0n,xu

1n,xu

2n,x.

Note that they can be readily derived also from the Lax pair (75) due to Lemma 5.2 and Lemma 5.1above.


By applying the manipulations similar to that fulfilled in the previous section by using the same change ofthe variables one can show that lattice (50) is reduced to the following finite field system which is of the form(3)

u0n+1,x = u0n,x −eu

0n

eu1n+ e

u0n+1

eu1n,

ujn+1,x = ujn,x +euj−1

n+1

eujn+1

− euj−1

n+1

eujn

− eujn

euj+1n

+ eujn+1

euj+1n

, 1 < j < N − 1,

uNn+1,x = uNn,x +euN−1n+1

euNn+1

− euN−1n+1

euNn

and its Lax pair is obtained from the system (57) by imposing additional conditions ϕ−1n = 0, ϕN+1

n = 0:

ϕ0n+1 = −e−u0

n+1+u0n

(

ϕ1n − ϕ0

n

)

+ ϕ1n,

ϕjn+1 = −e−u

j

n+1+uj

n

(

ϕj+1n − ϕj

n

)

+ ϕj+1n , 1 < j < N − 1,

ϕNn+1 = e−uN

n+1+uNn ϕN

n ,

ϕ0n,x = 0,

ϕjn,x = eu

j−1n −uj

n

(

ϕj−1n − ϕj

n

)

, 1 < j < N − 1,

ϕNn,x = eu

N−1n −uN

n

(

ϕN−1n − ϕN

n

)

.

We will investigate in more detail the case when N = 1. Then we obtain a system

{

u0n+1,x = u0n,x − eu0n−u1

n + eu0n+1−u1

n ,

u1n+1,x = u1n,x + eu0n+1−u1

n+1 − eu0n+1−u1

n

(76)

and its Lax pair

{

ϕ0n+1 = −e−u0

n+1+u0n

(

ϕ1n − ϕ0

n

)

+ ϕ1n,

ϕ1n+1 = e−u1

n+1+u1nϕ1

n,

{

ϕ0n,x = 0,

ϕ1n,x = eu

0n−u1

n

(

ϕ0n − ϕ1

n

)

.(77)

x-integrals and n-integrals of system (76) have the form

J1 =(

eu1n+1−u1

n − 1)(

eu0n+1−u0

n − 1)

;

J2 =(

eu1n−u1

n+1 − 1)(

eu0n+1−u0

n+2 − 1) (78)

14

and, respectively

I1 = u0n,x + u1n,x − eu0n−u1

n ;

I2 = u0n,xx + u0n,xu1n,x − eu

0n−u1

nu0n,x.(79)

Let us briefly discuss the methods of searching the integrals. Note that for some cases the Lax pairgives a convenient tool for solving the problem, however this way is applied not always. Therefore in theother cases we use algebraic method. We explain both approaches with the example of the system (76). Forconstructing the x-integrals we use the method of characteristic algebras and we use the Lax pair for findingthe n-integrals.

Assume that function H(u0n, u1n, u

0n+1, u

1n+1, . . .) is an x–integral for the system (76). Then according to

the definition the relationDxH(u0n, u

1n, u

0n+1, u

1n+1, . . .) = 0

should hold. Due to the chain rule equation (5.2) implies

K0H = 0,

where

K0 = u0n,x∂

∂u0n+ u1n,x

∂

∂u1n+ u0n+1,x

∂

∂u0n+1

+ u1n+1,x

∂

∂u1n+1

+ u0n−1,x

∂

∂u0n−1

+ u1n−1,x

∂

∂u1n−1

+ . . . .

We exclude the terms like u0n±i,x, u1n±i,x due to the system (76) and get

K0 =u0n,x∂

∂u0n+ u1n,x

∂

∂u1n+(

u0n,x − eu0n−u1

n + eu0n+1−u1

n

) ∂

∂u0n+1

+(

u1n,x + eu0n+1−u1

n+1 − eu0n+1−u1

n

) ∂

∂u1n+1

+ . . . .

Obviously equation K0H = 0 is overdetermined since the coefficients of the equation depend on u0n,x and u1n,xwhile the solution H does not depend on these variables, in other words we are interested on the solution Hof the equation (5.2) which solves in addition two more equations

X1H = 0 and X2H = 0,

where X1 = ∂∂u0

n,x, X2 = ∂

∂u1n,x

. More precisely we get a system of the first order linear partial differential

equations for one and the same unknown HK0H = 0,

X1H = 0,

X2H = 0.

(80)

Evidently operator K0 is represented as a linear combination of the vector fields

K0 = u0n,xY1 + u1n,xY2 +W,

where the coefficients of the operators Y1, Y2, W do not depend on the variables u0n,x, u1n,x:

Y1 =∂

∂u0n+

∂

∂u0n+1

+∂

∂u0n+2

,

Y2 =∂

∂u1n+

∂

∂u1n+1

,

W =(

eu0n+1−u1

n − eu0n−u1

n

) ∂

∂u0n+1

+(

eu0n+1−u1

n+1 − eu0n+1−u1

n

) ∂

∂u1n+1

+(

eu0n+2−u1

n+1 − eu0n+1−u1

n+1 + eu0n+1−u1

n − eu0n−u1

n

) ∂

∂u0n+2

.

Thus (80) is reduced to the formY1H = 0,

Y2H = 0,

WH = 0.

(81)

15

It is checked that system (81) is closed, i.e. all of the commutators [Y1, Y2], [Y1,W ] and [Y2,W ] are linearlyexpressed in terms of Y1, Y2, W such that:

[Y1, Y2] = 0,

[Y1,W ] =W,

[Y2,W ] = −W.

Since the system contains three equations then in order to get two functionally independent solutions welook for a solution depending on five variables

H = H(

u0n, u1n, u

0n+1, u

1n+1, u

0n+2

)

.

Then system (81) turns into

Hu0n+Hu0

n+1+Hu0

n+2= 0,

Hu1n+Hu1

n+1= 0,

(

eu0n+1−u1

n − eu0n−u1

n

)

Hu0n+1

+(

eu0n+1−u1

n+1 − eu0n+1−u1

n

)

Hu1n+1

+(

eu0n+2−u1

n+1 − eu0n+1−u1

n+1 + eu0n+1−u1

n − eu0n−u1

n

)

Hu0n+2

= 0.

(82)

In order to solve the system we have to reduce it due to Jacobi method to a normal (triangular) form

(

eu0n+1−u1

n − eu0n−u1

n

)

Hu0n−(

eu0n+1−u1

n+1 − eu0n+1−u1

n

)

Hu1n+1

−(

eu0n+2−u1

n+1 − eu0n+1−u1

n+1

)

Hu0n+2

= 0,

Hu1n+Hu1

n+1= 0,

(

eu0n+1−u1

n − eu0n−u1

n

)

Hu0n+1

+(

eu0n+1−u1

n+1 − eu0n+1−u1

n

)

Hu1n+1

+(

eu0n+2−u1

n+1 − eu0n+1−u1

n+1 + eu0n+1−u1

n − eu0n−u1

n

)

Hu0n+2

= 0.

Then by applying the consecutive integration according to the Jacobi algorithm we get the result (78).To construct the complete set of the n-integrals for the system (76) we use the Lax pair. At first we make

a change of the variables

ϕ0n = e−u0

nψ0n, ϕ1

n = e−u1nψ1

n

in the system (77) and get a new form of the Lax pair

Ψn,x = VnΨn, Ψn+1 = UnΨn, (83)

where Ψn =(

ψ0n, ψ

1n

)Tand

Vn =

(

u0n,x 0

1 u1n,x − eu0n−u1

n

)

, Un =

(

1 eu0n+1−u1

n − eu0n−u1

n

0 1

)

.

Now the Lax pair (83) satisfies all the requests of the Lemma 5.2. Therefore function r(k)N1,n is an n-integral.

In this case we have

r(2)21,n = u0n,x + u1n,x − eu

0n−u1

n ,

r(3)21,n = u0n,xx + 2u1n,xx + (u0n,x)

2 + u0n,xu1n,x + (u1n,x)

2 − 3eu0n−u1

nu0n,x + e2(u0n−u1

n).

After some slight simplifications we find integrals (79).


In equation (51) we put u0n = c0, uN+1n = cN :

u1n+1,x = u1n,x(u1

n+1)2

c0u2n,

ujn+1,x = ujn,x(uj

n+1)2

uj−1

n+1uj+1n

, 1 < j < N − 1,

uNn+1,x = uNn,x(uN

n+1)2

uN−1

n+1cN.

16

We will rewrite the Lax pair (58) with the above constraint for the field variables and additional constraintfor the eigenfunctions ϕ−1

n = 0, ϕN+1n = 0

ϕ0n+1 = − c0

u1n

(

ϕ0n − ϕ1

n

)

,

ϕjn+1 = −

uj

n+1

uj+1n

(

ϕjn − ϕj+1

n

)

, 1 < j < N − 1,

ϕNn+1 = −

uNn+1

cNϕNn ,

ϕ0n,x = 0,

ϕjn,x = −

ujn,x

ujn

(

ϕj−1n − ϕj

n

)

, 1 < j < N − 1,

ϕNn,x = −

uNn,x

uNn

(

ϕN−1n − ϕN

n

)

.

We put N = 1. Then we obtain an equation

u1n+1,x = u1n,x(

u1n+1

)2(84)

having the Lax pair

{

ϕ0n+1 = − 1

u1n

(

ϕ0n − ϕ1

n

)

,

ϕ1n+1 = −u1n+1ϕ

1n,

{

ϕ0n,x = 0,

ϕ1n,x = −

u1n,x

u1n

(

ϕ0n − ϕ1

n

)

.

x-integrals and n-integrals of equation (84) have the form

J = u1n +1

u1n+1

and, respectively

I =u1n,xxxu1n,x

−3

2

(u1n,xx)2

(u1n,x)2.


In the equation (52) we put u0n = c1, uN+1n = cN and obtain a system

u1n+1,x = u1n,x(u1

n+1−u2n)

c1−u1n

,

ujn+1,x = ujn,x(uj

n+1−uj+1

n )(uj−1

n+1−u

jn)

, 2 < j < N − 1,

uNn+1,x = uNn,xuNn+1−cN

(uN−1

n+1−uN

n )

with the Lax pair

ϕ0n+1 =

(

u1n − c1)

ϕ0n + ϕ1

n,

ϕjn+1 =

(

uj+1n − ujn+1

)

ϕjn + ϕj+1

n , 1 < j < N − 1,

ϕNn+1 =

(

cN − uNn+1

)

ϕNn ,

ϕ0n,x = 0,

ϕjn,x = −ujn,xϕ

j−1n , 1 < j < N − 1,

ϕNn,x = −uNn,xϕ

N−1n .

In the particular case N = 2 we get a system

u1n+1,x = u1n,x(u1

n+1−u2n)

c1−u1n

,

u2n+1,x = u2n,xu2n+1−c2

(u1n+1

−u2n)

(85)

with the Lax pair

ϕ0n+1 =

(

u1n − c1)

ϕ0n + ϕ1

n,

ϕ1n+1 =

(

u2n − u1n+1

)

ϕ1n + ϕ2

n,

ϕ2n+1 =

(

c2 − u2n+1

)

ϕ2n,

ϕ0n,x = 0,

ϕ1n,x = −u1n,xϕ

0n,

ϕ2n,x = −u2n,xϕ

1n.

(86)


J1 =(

u1n − c1) (

u2n+1 − c2) (

u2n − u1n+1

)

;

J2 = u1n+1u1n+2 − c1u

1n+2 − u1n+1u

2n+1 − c2u

2n + u2nu

2n+1

(87)

17

and, respectively

I1 =u1n,xxxu1n,x

+u2n,xxxu2n,x

−4

3

(

u1n,xxu1n,x

)2

−1

3

u1n,xxu2n,xx

u1n,xu2n,x

−4

3

(

u2n,xxu2n,x

)2

;

I2 =u1n,xxxxu1n,x

−13

3

u1n,xxu1n,xxx

(

u1n,x)2 −

2

3

u2n,xxu1n,xxx

u1n,xu2n,x

+2

3

u2n,xxu2n,xxx

(

u2n,x)2 +

1

3

u1n,xxu2n,xxx

u1n,xu2n,x

+32

9

(

u1n,xxu1n,x

)3

+

(

u1n,xx)2u2n,xx

(

u1n,x)2u2n,x

−2

3

u1n,xx(

u2n,xx)2

u1n,x(

u2n,x)2 −

8

9

(

u2n,xxu2n,x

)3

.

(88)

Now we briefly explain how the integrals (87) and (88) were constructed. Let us begin with x-integrals.The Lax pair can be written as

Ψn,x = VnΨn, Ψn+1 = UnΨn, (89)

where

Vn =

0 0 0−u1n,x 0 0

0 −u2n,x 0

, Un =

u1n − c1 1 00 u2n − u1n+1 10 0 c2 − u2n+1

.

Since the Lax pair (89) satisfies the conditions of Lemma 5.1, we can use it for deriving the x-integrals of thesystem (85). To this end we evaluate the product

P (k)n = Un+kUn+k−1 · · ·Un

and take its entry located at the right upper corner. It is easily checked that for k = 1 and k = 2 we gettrivial integrals J = 1 and J = c2 − c1. For k = 3 and k = 4 we obtain

p(3)13,n =

(

u1n+3 − c1) (

u1n+2 − c1)

+(

u2n+2 − c1) (

u1n+2 − c2)

+(

u2n+2 − c2) (

u2n+1 − c2)

,

p(4)13,n =

(

u1n+3 − c1) (

u1n+2 − c1) (

u1n+1 − c1)

−(

u1n+3 − c1) (

u2n+1 − c1) (

u1n+1 − c2)

−(

u1n+3 − u2n+2

) (

u1n+2 − u2n+1

) (

u1n+1 − c2)

+ (c2 − c1)(

u2n+1 − c2) (

u2n − c2)

.

After some slight simplifications we reduce them into integrals J1 and J2 given in (87).The next step is to explain the algebraic method of looking for the n-integrals. Let us emphasize that

this task is more difficult since the algebra Ln has a complicated structure.It is convenient to introduce new notations by setting un := u1n and vn := u2n. Thus we convert the system

(85) to the form{

un+1,x = F 1,

vn+1,x = F 2,(90)

where

F 1 = un,x(un+1 − vn)

c1 − un, F 2 = vn,x

vn+1 − c2(un+1 − vn)

.

At first we evaluate the subalgebra L(1)n , generated by the vector fields Y1,1, Y2,1, X1,1 and X2,1, where

Y1,1 =∂

∂un+D−1

n

(

∂F 1n

∂un+1

)

∂

∂un,x+D−1

n

(

∂F 2n

∂un+1

)

∂

∂vn,x

+D−1n

(

∂F 1n,x

∂un+1

)

∂

∂un,xx+D−1

n

(

∂F 2n,x

∂un+1

)

∂

∂vn,xx+ . . . ,

Y2,1 =∂

∂vn+D−1

n

(

∂F 1n

∂vn+1

)

∂

∂un,x+D−1

n

(

∂F 2n

∂vn+1

)

∂

∂vn,x

+D−1n

(

∂F 1n,x

∂vn+1

)

∂

∂un,xx+D−1

n

(

∂F 2n,x

∂vn+1

)

∂

∂vn,xx+ . . . ,

X1,1 =∂

∂un−1, X2,1 =

∂

∂vn−1.

Explicit form of the operators Y1,1 and Y2,1 are

Y1,1 =∂

∂un+

un,xun − vn−1

∂

∂un,x−

vn,xun − vn−1

∂

∂vn,x

18

+1

un − vn−1

(

un,xx +un,xvn,xvn − c2

)

∂

∂un,xx−

1

un − vn−1

(

vn,xx +2v2n,xvn − c2

)

∂

∂vn,xx+ . . . ,

Y2,1 =∂

∂vn+

vn,xvn − c2

∂

∂vn,x+

vn,xxvn − c2

∂

∂vn,xx+

vn,xxxvn − c2

∂

∂vn,xxx. . . .

It can be verified by analyzing the further terms in the series for Y1,1 that it is decomposed into a sum

Y1,1 = Y1 +1

un − vn−1Z1,

of two operators belonging the algebra L(1)n , since Z1 = (un − vn−1)

2[X2,1, Y1,1]. Here Y1 = ∂∂un

and thecoefficients in

Z1 = un,x∂

∂un,x− vn,x

∂

∂vn,x+

(

un,xx +un,xvn,xvn − c2

)

∂

∂un,xx−

(


)

∂

∂vn,xx+ . . .

do not depend on the variables un−1, vn−1. Therefore operators Y1, Z1 and Y2,1 commute with X1,1 and

X2,1. Moreover the operators X1,1, X2,1, Y1, Z1 and Y2,1 constitute a basis in the algebra L(1)n . Thus algebra

L(1)n is of dimension 5. However solution of the system

X1,1I = 0, X2,1I = 0, Y1I = 0, Z1I = 0, Y2,1I = 0

does not produce any n–integral for the system (90), since the relation DnI = I is not satisfied.

Therefore, we need to pass to the consideration of the second subalgebra L(2)n of the characteristic algebra

Ln generated by the operators

X1,1, X2,1, X1,2, X2,2, Y1,1, Y2,1, Y1,2, Y2,2.

In other words we add to L(1)n four extra operators

X1,2 =∂

∂un−2, X2,2 =

∂

∂vn−2, Y1,2 = D−1

n Y1,1Dn, Y2,2 = D−1n Y2,1Dn.

Two last operators are of the form

Y1,2 =un,x(vn−2 − c1)

(un−1 − vn−2)(un−1 − c1)

∂

∂un,x−

vn,xun−1 − vn−2

∂

∂vn,x

+1

un−1 − vn−2

(

un,xx(vn−2 − c1)

un−1 − c1+

2u2n,x(vn−2 − c1)

(un−1 − c1)(un − vn−1)+

un,xvn,x(un − c2)

(vn−1 − c2)(vn − c2)

)

∂

∂un,xx

−1

un−1 − vn−2

(

vn,xx +2v2n,x(un − c2)

(vn − c2)(vn−1 − c2)+

un,xvn,x(vn−2 − c1)

(un−1 − c1)(un − vn−1)

)

∂

∂vn,xx+ . . . ,

Y2,2 =−un,x

un − vn−1

∂

∂un,x+

vn,x(un − c2)

(un − vn−1)(vn−1 − c2)

∂

∂vn,x

−1

un − vn−1

(

un,xx +un,xvn,x(un − c2)

(vn − c2)(vn−1 − c2)

)

∂

∂un,xx

+un − c2

(un − vn−1)(vn−1 − c2)

(


)

∂

∂vn,xx+ . . . .

It can be proved that these operators are represented as linear combinations of the operators

Z2 = vn,x∂

∂vn,x−un,xvn,xvn − c2

∂

∂un,xx+

(


)

∂

∂vn,xx+ . . . ,

Y3 = un,x∂

∂un,x+ un,xx

∂

∂un,xx+ un,xxx

∂

∂un,xxx+ . . . ,

Z3 = 2u2n,x∂

∂un,xx− un,xvn,x

∂

∂vn,xx+

(

6un,xun,xx +3u2n,xvn,x

vn − c2

)

∂

∂un,xxx+ . . . ,

Z4 =3

2vn,x

∂

∂vn,xx+

(

un,xx −un,xvn,xxvn,x

+3

2

un,xvn,xvn − c2

)

∂

∂un,xxx+ . . . ,

19

Y4 = 3un,xvn,x∂

∂un,xxx− 3v2n,x

∂

∂vn,xxx+ (14vn,xun,xx + 4un,xvn,xx)

∂

∂un,xxxx+ . . .

which are simpler, since they commute with X1,1, X1,2, X2,1, X2,2.We prove that system of equations

Y1I = 0, Y2,1I = 0, Y3I = 0, Z2I = 0, Z3I = 0, Z4I = 0, Y4I = 0 (91)

is closed or in other words algebra L(2)n is of finite dimension. We look for the solution of (91) in the form

I = I (un, vn, un,x, vn,x, un,xx, vn,xx, un,xxx, vn,xxx, un,xxxx) .

We reduce it to normal form and solve by Jacobi method. As a result we get (88), where one has to replaceun := u1n and vn := u2n.



u1n+1,x = u1n,xu1n+1(u

1n+1−u2

n)u2n(c0−u1

n),

ujn+1,x = ujn,xuj

n+1(uj

n+1−uj+1

n )uj+1n (uj−1

n+1−u

jn)

, 2 < j < N − 1,

uNn+1,x = uNn,xuNn+1(u

Nn+1−cN)

cN(uN−1

n+1−uN

n ).

We rewrite the Lax pair (60) with the above restrictions and by setting the conditions ϕ−1n = 0, ϕN+1

n = 0

ϕ0n+1 =

(

1− c0u1n

)

ϕ0n − c0

u1nϕ1n,

ϕjn+1 =

(

1−uj

n+1

uj+1n

)

ϕjn −

uj

n+1

uj+1n

ϕj+1n , 1 < j < N − 1,

ϕNn+1 =

(

1−uNn+1

cN

)

ϕNn ,

ϕ0n,x = 0,

ϕjn,x = ϕj

n,x =ujn,x

ujn

(

ϕj−1n + ϕj

n

)

, 1 < j < N − 1,

ϕNn,x =

uNn,x

uNn

(

ϕN−1n + ϕN

n

)

.

We concentrate on a simplest case by taking N = 1. Then we get an equation

u1n+1,x = u1n,xu1n+1

(

u1n+1 − c1)

c1 (c0 − u1n), (92)

and its Lax pair

ϕ0n+1 =

(

1− c0u1n

)

ϕ0n − c0

u1nϕ1n,

ϕ1n+1 =

(

1−u1n+1

c1

)

ϕ1n,

{

ϕ0n,x = 0,

ϕ1n,x =

u1n,x

u1n

(

ϕ0n + ϕ1

n

)

.


J =u1nu1n+1

+c0u1n+1

+u1nc1

and, respectively

I =u1n,xxxu1n,x

−3

2

(u1n,xx)2

(u1n,x)2.



u1n+1,x = u1n,x(c0−u1

n+1)(u1n+1−u2

n)(c0−u1

n)(u1n−u2

n),

ujn+1,x = ujn,x(uj−1

n+1−u

j

n+1)(uj

n+1−uj+1

n )(uj−1

n+1−u

jn)(uj

n−uj+1n )

, 2 < j < N − 1,

uNn+1,x = uNn,x(uN−1

n+1−uN

n+1)(uNn+1−cN)

(uN−1

n+1−uN

n )(uNn −cN )

.

20

We rewrite the Lax pair (61) with the constraint u0n = c0, uN+1n = cN , ϕ−1

n = 0, ϕN+1n = 0

ϕ0n+1 =

c0−u1n

c0−u1nϕ0n +

(

1−c0−u1

n

c0−u1n

)

ϕ1n,

ϕjn+1 =

uj

n+1−uj+1

n

ujn−u

j+1n

ϕjn +

(

1−uj

n+1−uj+1

n

ujn−u

j+1n

)

ϕj+1n , 1 < j < N − 1,

ϕNn+1 =

uNn+1−cN

uNn −cN

ϕNn ,

ϕ0n,x = 0,

ϕjn,x =

ujn,x

uj−1n −u

jn

(

ϕj−1n + ϕj

n

)

, 1 < j < N − 1,

ϕNn,x =

uNn,x

uN−1n −uN

n

(

ϕN−1n + ϕN

n

)

,

By putting N = 2 we arrive at a system

u1n+1,x = u1n,x(c0−u1

n+1)(u1n+1−u2

n)(c0−u1

n)(u1n−u2

n),

u2n+1,x = u2n,x(u1

n+1−u2n+1)(u

2n+1−cN)

(u1n+1

−u2n)(u2

n−c2).

(93)

admitting the Lax pair

ϕ0n+1 =

c0−u1n

c0−u1nϕ0n +

(

1−c0−u1

n

c0−u1n

)

ϕ1n,

ϕ1n+1 =

u1n+1−u2

n

u1n−u2

nϕ1n +

(

1−u1n+1−u2

n

u1n−u2

n

)

ϕ2n,

ϕ2n+1 =

u2n+1−c2

u2n−c2

ϕ2n,

ϕ0n,x = 0,

ϕ1n,x =

u1n,x

u0n−u1

n

(

ϕ0n + ϕ1

n

)

,

ϕ2n,x =

u2n,x

u1n−u2

n

(

ϕ1n + ϕ2

n

)

.


J1 =

(

u1n − c0) (

u2n − c2) (

u1n+1 − u2n)

(

u1n+1 − u1n) (

u2n+1 − u2n) ;

J2 =

(

u1n+2 − c0) (

u2n − c2) (

u2n+1 − u1n+1

) (

u2n+2 − u2n+1

)

(

u1n+1 − c0) (

u2n+2 − c2) (

u2n+1 − u2n) (

u1n+2 − u2n+1

)

and, respectively

I1 =u1n,xu

2n,x

(u1n − c0) (u2n − c2) (u1n − u2n);

I2 =u1n,xxu1n,x

−2u1n,xu1n − c0

+u2n,x

(

u1n − c2)

(u2n − c2) (u1n − u2n).


In the equations (55), (62) we put u0n = c0, uN+1n = cN , ϕ−1

n = 0, ϕN+1n = 0 and get a system

u1n+1,x = u1n,xsinh(c0−u1

n+1) sinh(u1n+1−u2

n)sinh(c0−u1

n) sinh(u1n−u2

n),

ujn+1,x = ujn,xsinh(uj−1

n+1−u

j

n+1) sinh(uj

n+1−uj+1

n )sinh(uj−1

n+1−u

jn) sinh(uj

n−uj+1n )

, 1 < j < N − 1,

uNn+1,x = uNn,xsinh(uN−1

n+1−uN

n+1) sinh(uNn+1−cN)

sinh(uN−1

n+1−uN

n ) sinh(uNn −cN )

.

with the Lax pair

ϕ0n+1 = e

2(c0−u1n)−1

e2(c0−u1n)−1

ϕ0n +

(

1− e2(c0−u1

n)−1

e2(c0−u1n)−1

)

ϕ1n,

ϕjn+1 = e

2(ujn+1

−uj+1n)−1

e2(uj

n−uj+1n )

−1

ϕjn +

(

1− e2(uj

n+1−u

j+1n )

−1

e2(uj

n−uj+1n )

−1

)

ϕj+1n , 1 < j < N − 1,

ϕNn+1 = e

2(uNn+1

−cN)−1

e2(uNn −cN )−1

ϕNn ,

ϕ0n,x = 0,

ϕjn,x =

2ujn,x

e2(uj−1

n −ujn)

−1

(

ϕj−1n − ϕj

n

)

, 1 < j < N − 1,

ϕNn,x =

2uNn,x

e2(uN−1

n −uNn )−1

(

ϕN−1n − ϕN

n

)

.

21

We put N = 1 and obtain an equation

u1n+1,x = u1n,xsinh

(

c0 − u1n+1

)

sinh(

u1n+1 − c1)

sinh (c0 − u1n) sinh (u1n − c1)

, (94)

having the Lax pair

ϕ0n+1 = e

2(c0−u1n)−1

e2(c0−u1n)−1

ϕ0n +

(

1− e2(c0−u1

n)−1

e2(c0−u1n)−1

)

ϕ1n,

ϕ1n+1 = e

2(u1n+1

−c1)−1

e2(u1n−c1)−1

ϕ1n,

ϕ0n,x = 0,

ϕ1n,x =

2u1n,x

e2(c0−u1n)−1

(

ϕ0n − ϕ1

n

)

.


J =1

sinh(c0 − c1)

[

arctanh

(

(cosh(c0 − c1) + 1) tanh( c0+c12 − u1n)

sinh(c0 − c1)

)

−arctanh

(

(cosh(c0 − c1) + 1) tanh( c0+c12 − u1n+1)

sinh(c0 − c1)

)]

and, respectively

I =u1n,x

sinh(c0 − u1n) sinh(c1 − u1n).

5.8 Scalar equations

Here we present a list of Darboux integrable scalar equations obtained from the equations (51)–(55) byimposing degenerate boundary conditions:

1) un+1,x = un,x(un+1)2;

2) un+1,x = un,xc− un+1

un;

3) un+1,x = un,x(un+1 + c)(un+1 + c1)

un;

4) un+1,x = un,xun+1(un+1 − c)

un(un − c);

5) un+1,x = un,xsinh(un+1) sinh(un+1 − c)

sinh(un) sinh(un − c).

Apparently they are new.

Conclusions

The paper proposes an algebraic method for the classification of integrable cases of differential-differenceequations of the form (1). It is shown that all known integrable equations of this form given in the listpresented in [17] admit reductions as finite systems of differential-difference equations integrable in sense ofDarboux. We used this property of the equation (1) as the basis for the classification criterion. It is convenientto formalize the property of Darboux integrability in terms of its characteristic Lie-Rinehart algebras. Thearticle gives a definition of these algebras for systems of differential-difference equations and investigates theirbasic properties that may be needed in solving the classification problem.

Funding

One of the authors A.R. Khakimova was supported in part by Young Russian Mathematics award.

References

[1] Bogdanov, L.V. “Dunajski-Tod equation and reductions of the generalized dispersionless 2DTL hierar-chy.” Phys. Lett. A 376, no. 45 (2012): 2894–2898.

22

[2] Bogdanov, L.V. and B.G. Konopelchenko. “On dispersionless BKP hierarchy and its reductions.” J.Nonlinear Math. Phys. 12, suppl. 1 (2005): 64–73.

[3] Calderbank, D.M.J. “Integrable background geometries.” SIGMA 10, no. 034 (2014): 51 pp.

[4] Calderbank, D.M.J. and B. Kruglikov. “Integrability via geometry: dispersionless differential equations inthree and four dimensions.” Commun. Math. Phys. (2020). https://doi.org/10.1007/s00220-020-03913-y

[5] Clery, F. and E.V. Ferapontov. “Dispersionless Hirota equations and the genus 3 hyper elliptic divisor.”Commun. Math. Phys. 376, no. 2 (2019): 1397–1412.

[6] Darboux, G. Lecons sur la theorie generale des surfaces et les applications geometriques du calcul in-finitesimal. Paris: Gauthier-Villars, 1896.

[7] Doubrov, B., E.V. Ferapontov, B. Kruglikov, and V.S. Novikov. “On integrability in Grassmann ge-ometries: integrable systems associated with fourfolds in Gr(3, 5).” Proc. London Math. Soc. 116, no. 5(2018): 1269–1300.

[8] Doubrov, B., E.V. Ferapontov, B. Kruglikov, and V.S. Novikov. “Integrable systems in four dimensionsassociated with six–folds in Gr(4, 6).” Int. Math. Res. Not. IMRN 2019, no. 21 (2019): 6585–6613.

[9] Dunajski, M., L.J. Mason, and P. Tod. “Einstein-Weyl geometry, the dKP equation and twistor theory.”J. Geom. Phys. 37, no. 1-2 (2001): 63–93.

[10] Ferapontov, E.V., I.T. Habibullin, M.N. Kuznetsova, and V.S. Novikov. “On a class of 2D integrablelattice equations.” J. Math. Phys. 61, no. 7 (2020): 073505, 15 pp.

[11] Ferapontov, E.V., L. Hadjikos, and K.R. Khusnutdinova. “Integrable equations of the dispersionlessHirota type and hypersurfaces in the Lagrangian Grassmannian.” Int. Math. Res. Not. IMRN 2010, no.3 (2010): 496–535.

[12] Ferapontov, E.V. and K.R. Khusnutdinova. “Hydrodynamic reductions of multidimensional dispersion-less PDEs: the test for integrability.” J. Math. Phys. 45, no. 6 (2004): 2365–2377.

[13] Ferapontov, E.V. and B.S. Kruglikov. “Dispersionless integrable systems in 3D and Einstein-Weyl ge-ometry”, J. Differ. Geom. 97, no. 2 (2014): 215–254.

[14] Ferapontov, E.V. and A. Moro. “Dispersive deformations of hydrodynamic reductions of 2D dispersionlessintegrable systems.” J. Phys. A: Math. Theor. 42 (2009) 035211, 15 pp.

[15] Ferapontov, E.V., A. Moro, and V. S. Novikov. “Integrable equations in 2+1 dimensions: deformationsof dispersionless limits.” J. Phys. A: Math. Theor. 42 (2009) 18 pp.

[16] Ferapontov, E.V. and A.V. Odesskii. “Integrable Lagrangians and modular forms.” J. Geom. Phys. 60,no. 6–8 (2010): 896–906.

[17] Ferapontov, E.V., V.S. Novikov, and I. Roustemoglou. “On the classification of discrete Hirota-typeequations in 3D.” Int. Math. Res. Not. IMRN 2015, no. 13 (2015): 4933–4974.

[18] Ganzha, E.I. and S.P. Tsarev. Integration of Classical Series An, Bn, Cn, of Exponential Systems.Krasnoyarsk: Krasnoyarsk State Pedagogical University Press, 2001.

[19] Gibbons, J. and Y. Kodama. “A method for solving the dispersionless KP hierarchy and its exactsolutions. II.” Phys. Lett. A 135, no. 3 (1989): 167–170.

[20] Gibbons, J. and S. P. Tsarev. “Reductions of the Benney equations.” Phys. Lett. A 211, no. 3 (1996):19–24. “Conformal maps and reductions of the Benney equations.” Phys. Lett. A 258 (1999): 263–270.

[21] Goursat, M.E. “Recherches sur quelques equations aux derivees partielles du second ordre.” Annales dela Facult? des sciences de Toulouse: Math?matiques 1, no. 2 (1899): 31–78, 439–463.

[22] Habibullin, I. “Characteristic Lie rings, finitely-generated modules and integrability conditions for (2+1)-dimensional lattices.” Phys. Scr. 87, no. 6 (2013): 065005.

[23] Habibullin, I.T. and M.N. Kuznetsova. “A classification algorithm for integrable two-dimensional latticesvia Lie-Rinehart algebras.” Theoret. and Math. Phys. 203, no. 1 (2020): 569–581.

[24] Habibullin, I.T., M.N. Kuznetsova, and A.U. Sakieva. “Integrability conditions for two-dimensional Toda-like equations.” J. Phys. A: Math. Theor. 53, no. 39 (2020): 395203 , 25 pp.

[25] Habibullin, I., A. Pekcan, and N. Zheltukhina. “On Some Algebraic Properties of Semi-Discrete Hyper-bolic Type Equations.” Turkish J. Math. 32, no. 3 (2008): 277–292.

[26] Habibullin, I.T. and M.N. Poptsova. “Classification of a Subclass of Two-Dimensional Lattices via Char-acteristic Lie Rings.” SIGMA 13, no. 073 (2017): 26 pp.

[27] Habibullin, I., N. Zheltukhina, and A. Pekcan.“On the classification of Darboux integrable chains.” J.Math. Phys. 49, no. 10 (2008): 1–39.

23

[28] Habibullin, I., N. Zheltukhina, and A. Pekcan. “Complete list of Darboux integrable chains of the formt1,x = tx + d(t, t1).” J. Math. Phys. 50, no. 10 (2009): 1–23.

[29] Habibullin, I., N. Zheltukhina, and A. Sakieva. “On Darboux-integrable semi-discrete chains.” J. Phys.A: Math. Theor. 43, no. 43 (2010): 434017, 14 pp.

[30] Habibullin, I., K. Zheltukhin, and M. Yangubaeva. “Cartan matrices and integrable lattice Toda fieldequations”, J. Phys. A: Math. Theor. 44, no. 46 (2011): 465202.

[31] Hitchin, N.J. “Complex manifolds and Einstein’s equations, Twistor geometry and nonlinear systems.”Lecture Notes in Math. 970, (1982): 73–99.

[32] Krichever, I.M. “The τ -function of the universal whitham hierarchy, matrix models and topological fieldtheories.” Comm. Pure Appl. Math. 47, no. 4 (1994): 437–475.

[33] Leznov, A.N. and M.V. Savel’ev. Group Methods of Integration of Nonlinear Dynamical Systems. Nauka:Moscow (in Russian), 1985.

[34] Mikhailov A.V., M.A. Olshanetsky, and A.M. Perelomov. “Two-dimensional generalized Toda lattice.”Commun. Math. Phys. 79, no. 4 (1981): 473–488.

[35] Odesskii, A.V. and V.V. Sokolov. “Integrable pseudopotentials related to generalized hypergeometricfunctions.” Sel. Math. New Ser. 16, (2010): 145–172.

[36] Pavlov, M.V. “New integrable (2+1)-equations of hydrodynamic type.” Russian Math. Surv. 58, no. 2(2003): 386–387.

[37] Pavlov, M.V. “Classifying integrable egoroff hydrodynamic chains.” Theoret. and Math. Phys. 138, no.1 (2004): 45–58.

[38] Penrose, R. “Nonlinear gravitons and curved twistor theory.” Gen. Relat. Gravit. 7, no. 1 (1976): 31–52.

[39] Poptsova, M.N. and I.T. Habibullin. “Algebraic properties of quasilinear two-dimensional lattices con-nected with integrability.” Ufa Math. J. 10, no. 3 (2018): 86–105.

[40] Rinehart, G. “Differential forms for general commutative algebras.” Trans. Amer. Math. Soc. 108 (1963):195–222.

[41] Shabat, A.B. and R.I. Yamilov. “Exponential Systems of Type I and the Cartan Matrices.” Preprint,Soviet Academy of Sciences, Bashkirian Branch, Ufa (in Russian) (1981): 22 pp.

[42] Smirnov, S.V. “Darboux integrability of discrete two-dimensional Toda lattices.” Theoret. and Math.Phys. 182, no. 2 (2015): 189–210.

[43] Voronova, Yu.G. and A.V. Zhiber. “Symmetries and Goursat problem for system of equationsuxy = eu + vuy, vxy = −eu + vvy”, Ufa Math. J. 5, no. 3 (2013): 20–27.

[44] Ward, R.S. “Einstein-Weyl spaces and SU (infinity) Toda fields.” Class. Quantum Grav. 7, no. 4 (1990):95–98.

[45] Zakharov, V.E. “Dispersionless limit of integrable systems in 2+1 dimensions.” Singular Limits of Dis-persive Waves, NATO ASI series 320, (1994): 165–174.

[46] Zhiber, A.V. and O.S. Kostrigina. “Exactly integrable models of wave processes.” Vestnik UGATU (inRussian) 9, no. 7(25) (2007): 83–89.

[47] Zhiber, A.V. and V.V. Sokolov. “Exactly integrable hyperbolic equations of Liouville type.” RussianMath. Surveys 56, no. 1 (2001): 61–101.

24

Date post:	03-Apr-2022
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Characteristic Lie Algebras of Integrable Diﬀerential ...

Documents