Algebraic Properties of Quasilinear Two-Dimensional Lattices connected with integrability
aa r X i v : . [ n li n . S I] A ug Algebraic Properties of QuasilinearTwo-Dimensional Lattices connected withintegrability
I.T. Habibullin , , M.N. Kuznetsova Institute of Mathematics, Ufa Federal Research Centre, Russian Academy ofSciences, 112, Chernyshevsky Street, Ufa 450008, Russian Federation Bashkir State University, 32 Validy Street, Ufa 450076 , Russian FederationE-mail: [email protected] , [email protected] Abstract.
In the article a classification method for nonlinear integrable equations with threeindependent variables is discussed based on the notion of the integrable reductions.We call the equation integrable if it admits a large class of reductions being Darbouxintegrable systems of hyperbolic type equations with two independent variables.The most natural and convenient object to be studied within the frame of thisscheme is the class of two dimensional lattices generalizing the well-known Todalattice. In the present article we deal with the quasilinear lattices of the form u n,xy = α ( u n +1 , u n , u n − ) u n,x u n,y + β ( u n +1 , u n , u n − ) u n,x + γ ( u n +1 , u n , u n − ) u n,y + δ ( u n +1 , u n , u n − ). We specify the coefficients of the lattice assuming that there existcutting off conditions which reduce the lattice to a Darboux integrable hyperbolic typesystem of the arbitrarily high order. Under some extra assumption of nondegeneracywe described the class of the lattices integrable in the sense indicated above. Thereare new examples in the obtained list of chains. lgebraic Properties of Quasilinear Two-Dimensional Lattices
1. Introduction
Integrable equations with three independent variables have a wide range of applicationsin physics. It suffices to recall such well-known nonlinear models as the KP equation,the Davey-Stewartson equation, the Toda lattice equation, and so on. From the point ofview of integration and classification, multidimensional equations are the most complex.Different approaches to study the integrable multidimensional models are discussed,for example, in the papers [1]–[9]. It is known that the symmetry approach [10, 11],which has proved to be a very effective method for classifying integrable equations in1 + 1 dimensions, is not so effective in the multidimensionality [12]. For studyingmultidimensional equations, the idea of the reduction is often used, when the researchesreplace the equation with a system of equations with fewer independent variables. Theexistence of a wide class of integrable reductions with two independent variables, as arule, indicates the integrability of an equation with three independent variables. Amongthe specialists, the most popular method is the method of hydrodynamic reductions,when the presence of an infinite set of integrable systems of hydrodynamic type is takenas a sign of integrability of the equation, the general solution of each of which generatessome solution of the equation under consideration (see, for example, [13, 1, 2]). Thehistory of the method and related references can be found in the survey [3].In our works [14, 15] we use an alternative approach. We call this equationintegrable if it admits an infinite class of reductions in the form of Darboux-integrablesystems of partial differential equations of hyperbolic type with two independentvariables. In solving classification problems for multidimensional equations, theapparatus of characteristic Lie algebras can be used in this formulation (a detailedexposition can be found in [17, 18]). This direction in the theory of integrability seemsto us promising. Consider a nonlinear chain u n,xy = f ( u n +1 , u n , u n − , u n,x , u n,y ) (1.1)with three independent variables, where the sought function u = u n ( x, y ) depends onthe real x , y , and the integer n . For the chain (1.1), the desired finite-field reductionsare obtained in a natural way, in a sufficiently suitable way to break off the chain attwo integer points u N = ϕ ( x, y, u N +1 , ... ) , (1.2) u n,xy = f ( u n +1 , u n , u n − , u n,x , u n,y ) , N < n < N , (1.3) u N = ϕ ( x, y, u N − , ... ) . (1.4)Examples of such boundary conditions can be found below (see (4.29), (4.30)). Thefollowing two very significant circumstances should be noted:i) for any known integrable chain of the form (1.1) there are cut-off conditions reducingit to a Darboux-integrable system of the form (1.2)-(1.4) of arbitrarily large order N = N − N − lgebraic Properties of Quasilinear Two-Dimensional Lattices ϕ , ϕ , and f is constructively determined by therequirement of integrability of the system in the sense of Darboux.These two facts serve as motivation for the following definition (see also the work[14]): Definition 1
A chain (1.1) is called integrable if there exist functions ϕ and ϕ suchthat for any choice of a pair of integers N , N , where N < N − , the hyperbolic typesystem (1.2)-(1.4) is Darboux integrable. In the present paper we investigate quasilinear chains of the following form u n,xy = αu n,x u n,y + βu n,x + γu n,y + δ, (1.5)assuming that the functions α = α ( u n +1 , u n , u n − ), β = β ( u n +1 , u n , u n − ), γ = γ ( u n +1 , u n , u n − ), δ = δ ( u n +1 , u n , u n − ) are analytic in the domain D ⊂ C . Wealso assume that the derivatives ∂α ( u n +1 , u n , u n − ) ∂u n +1 ∂α ( u n +1 , u n , u n − ) ∂u n − (1.6)differ from zero.The main result of this paper is the proof of the following assertion Theorem 1
The quasilinear chain (1.5), (1.6) is integrable in the sense of Definition1 if and only if it is reduced by point transformations to one of the following forms i ) u n,xy = α n u n,x u n,y ,ii ) u n,xy = α n ( u n,x u n,y − u n ( u n,x + u n,y ) + u n ) + u n,x + u n,y − u n ,iii ) u n,xy = α n ( u n,x u n,y − s n ( u n,x + u n,y ) + s n ) + s ′ n ( u n,x + u n,y − s n ) , where s n = u n + C, s ′ n = 2 u n , α n = 1 u n − u n − − u n +1 − u n ,C is an arbitrary constant. We note that equation i) was found earlier in the papers [27], [28] by Ferapontov andShabat and Yamilov, equations ii) and iii) appear to be new. By applying additionalconditions of the form x = ± y to the equations i)-iii), we obtain 1 + 1 -dimensionalintegrable chains. It is easily verified that by point transformations they are reduced tothe equations found earlier by Yamilov (see [29]).Following Definition 1, we suppose that there are cut-off conditions such that byimposing them at two arbitrary points n = N , n = N ( N < N −
1) to the chain (1.5)we obtain a system of hyperbolic type equations u N = ϕ ,u n,xy = α n u n,x u n,y + β n u n,x + γ n u n,y + δ n , N < n < N , (1.7) u N = ϕ . that is integrable in the sense of Darboux. lgebraic Properties of Quasilinear Two-Dimensional Lattices x − and y − integrals. A function I that depends on a finite set of dynamical variables u , u x , u y , . . . is called a y -integral if it satisfies the equation D y I = 0, where D y is theoperator of total derivative with respect to the variable y , and the vector u has thecoordinates u N +1 , u N +2 , . . . , u N − . Since the system (1.7) is autonomous, we consideronly autonomous nontrivial integrals. It can be shown that the y − integral does notdepend on u y , u yy , . . . . Therefore, we will consider only y − integrals that depend on atleast one dynamic variable u , u x , . . . . We note that nowadays the Darboux integrablediscrete and continuous models are intensively studied (see, [14, 17], [19]-[26]).We give one more argument in favor of our Definition 1 concerning the integrabilityproperty of a two-dimensional chain. The problem of finding a general solution ofa Darboux-integrable system reduces to the problem of solving a system of ordinarydifferential equations. Usually these ODEs are solved explicitly. On the other hand,any solution of the considered hyperbolic system (1.7) easily extends beyond the interval[ N , N ] and generates the solution of the corresponding chain (1.5). Therefore, in thiscase the chain (1.5) has a large set of exact solutions.Let us briefly explain the structure of the paper. In § § α , β , γ . Paragraph 4 is devoted to thesearch for the function δ . Here we also give the final form of the desired chain (4.28)that is integrable in the sense of Definition 1 and the proof of the Theorem 1 is given.
2. Characteristic Lie algebras
Since the chain (1.5) is invariant under the shift of the variable n , then without loss ofgenerality we can put N = −
1. In what follows we consider a system of hyperbolicequations u − = ϕ ,u n,xy = α n u n,x u n,y + β n u n,x + γ n u n,y + δ n , ≤ n ≤ N, (2.1) u N +1 = ϕ . Recall that here α n = α ( u n − , u n , u n +1 ), β n = β ( u n − , u n , u n +1 ), γ n = γ ( u n − , u n , u n +1 ), δ n = δ ( u n − , u n , u n +1 ). Suppose that the system (2.1) is Darboux integrable and that I ( u , u x , . . . ) is its nontrivial y -integral. The latter means that the function I mustsatisfy the equation D y I = 0, where D y is the operator of total derivative with respectto the variable y . The operator D y acts on the class of functions of the form I ( u , u x , . . . )due to the rule D y I = Y I , where Y = N X i =0 (cid:18) u i,y ∂∂u i + f i ∂∂u i,x + f i,x ∂∂u i,xx + · · · (cid:19) . (2.2) lgebraic Properties of Quasilinear Two-Dimensional Lattices f i = α i u i,x u i,y + β i u i,x + γ i u i,y + δ i is the right hand side of the lattice (1.5). Hence,the function I satisfies the equation Y I = 0. The coefficients of the equation
Y I = 0depend on the variables u i,y , while its solution I does not depend on u i,y , therefore thefunction I actually satisfies the system linear equations: Y I = 0 , X j I = 0 , j = 1 , . . . , N, (2.3)where X i = ∂∂u i,y . It follows from (2.3) that the commutator Y i = [ X i , Y ] of the operators Y and X i for i = 0 , , ...N also annuls I . We use the explicit coordinate representationof the operator Y i : Y i = ∂∂u i + X i ( f i ) ∂∂u i,x + X i ( D x f i ) ∂∂u i,xx + · · · (2.4)By the special form of the function f i , the operator Y can be represented in the form: Y = N X i =0 u i,y Y i + R, (2.5)where R = N X i =0 ( f i − u i,y X i ( f i )) ∂∂u i,x + ( f i,x − u i,y X i ( D x f i )) ∂∂u i,xx + · · · == N X i =0 ( β i u i,x + δ i ) ∂∂u i,x ++ (( α i u i,x + γ i )( β i u i,x + δ i ) + D x ( β i u i,x + δ i )) ∂∂u i,xx + · · · (2.6)Denote by F the ring of locally analytic functions of the dynamical variables u , u x , u y , . . . .Consider the Lie algebra L ( y, N ) over the ring F generated by the differentialoperators Y, Y , Y , ..., Y N . It is clear that the operations of computing the commutatorof two vector fields and multiplying the vector field by a function satisfy the followingconditions: [ Z, gW ] = Z ( g ) W + g [ Z, W ] , (2.7)( gZ ) h = gZ ( h ) , (2.8)where Z, W ∈ L ( y, N ), g, h ∈ F . Consequently, the pair ( F , L ( y , N )) has the structureof the Lie-Rinehart algebra ‡ (see [30]). We call this algebra the characteristic Lie algebraof the system of equations (2.1) along the direction y . It is well known (see [20, 17])that the function I is a y -integral of the system (2.1) if and only if it belongs tothe kernel of each operator from L ( y, N ). Since the y -integral depends only on a finitenumber of dynamic variables, we can use the well-known Jacobi theorem on the existenceof a nontrivial solution of a system of first-order linear differential equations with oneunknown function. From this theorem it is easy to deduce that in the Darboux integrablecase in the algebra L ( y, N ) there must exist a finite basis Z , Z , ...Z k , consisting of ‡ We thank D.V. Millionshchikov who drew our attention to this circumstance. lgebraic Properties of Quasilinear Two-Dimensional Lattices Z of L ( y, N ) can be representedas a linear combination Z = a Z + a Z + ...a k Z k , where the coefficients a , a , ...a k areanalytic functions of dynamical variables defined in some open set. Moreover, from theequality a Z + a Z + ...a k Z k = 0 it follows that a = a = ... = a k = 0. In this case, wecall the algebra L ( y, N ) finite-dimensional. Similarly, we can define the characteristicalgebra L ( x, N ) in the direction x . It is clear that the system (2.1) is Darboux integrableif and only if the characteristic algebras in both directions are finite-dimensional.For the sake of convenience, we introduce the notation ad X ( Z ) := [ X, Z ]. Wenote that in our study the operator ad D x plays a key role. Below we shall apply D x tosmooth functions depending on dynamical variables u , u x , u xx , . . . . As was shown above,the operators D y and Y coincide on this class of functions. Therefore, the equality[ D x , D y ] = 0 immediately implies [ D x , Y ] = 0. Replacing Y by virtue of (2.5), andcollecting in the resulting relation the coefficients of the independent variables { u i,y } Ni =0 ,we obtain [ D x , Y i ] = − a i Y i , a i = α i u i,x + γ i . (2.9)It is clear that the operator ad D x takes the characteristic Lie algebra into itself. Wedescribe the kernel of this mapping: Lemma 1 [16, 18, 17] If the vector field of the form Z = X i z ,i ∂∂u i,x + z ,i ∂∂u i,xx + · · · (2.10) solves the equation [ D x , Z ] = 0 , then Z = 0 .
3. Method of test sequences
We call the sequence of operators W , W , W , . . . in the algebra L ( y, N ) a test sequenceif ∀ m holds: [ D x , W m ] = m X j =0 w j,m W j . (3.1)The test sequence allows us to derive the integrability conditions for a system ofhyperbolic type (2.1) (see [19, 17, 20]). Indeed, assume that (2.1) is Darbouxintegrable. Then among the operators W , W , W , . . . there is only a finite set of linearlyindependent elements through which all the others are expressed. In other words, thereexists an integer k such that the operators W , . . . W k are linearly independent and W k +1 is expressed as follows: W k +1 = λ k W k + · · · + λ W . (3.2)We apply the operator ad D x to both sides of the equality (3.2). As a result, we obtainthe relation k X j =0 w j,k +1 W j + w k +1 ,k +1 k X j =0 λ j W j = k X j =0 D x ( λ j ) W j + lgebraic Properties of Quasilinear Two-Dimensional Lattices λ k k X j =0 w j,k W j + λ k − k − X j =0 w j,k − W j + · · · + λ w , W . (3.3)Collecting coefficients for independent operators, we obtain a system of differentialequations for the coefficients λ , λ , . . . λ k . The resulting system is overdetermined, since λ j is a function of a finite number of dynamical variables u , u x , . . . . The consistencyconditions for this system define the integrability conditions for (2.1). For example,collecting the coefficients for W k we get the first equation of the indicated system: D x ( λ k ) = λ k ( w k +1 ,k +1 − w k,k ) + w k,k +1 , (3.4)which is also overdetermined.Below in this section, we use two test sequences to refine the form of the functions α n , β n , γ n . Let us define a sequence of operators in the characteristic algebra L ( y, N ) by thefollowing recurrence formula: Y , Y , W = [ Y , Y ] , W = [ Y , W ] , . . . W k +1 = [ Y , W k ] , . . . (3.5)Above (see (2.9), the commutation relations for the first two terms of this sequence werederived:[ D x , Y ] = − a Y = − ( α u ,x + γ ) Y , [ D x , Y ] = − a Y = − ( α u ,x + γ ) Y . (3.6)Applying the Jacobi identity and using the last formulas, we derive:[ D x , W ] = − ( a + a ) W − Y ( a ) Y + Y ( a ) Y . (3.7)We can prove by induction that (3.5) is a test sequence. Moreover, for any k ≥ D x , W k ] = p k W k + q k W k − + · · · , (3.8)holds where the functions p k , q k are found due to the rule p k = − ( a + ka ) , q k = k − k Y ( a ) − Y ( a ) k. (3.9)By assumption, in the algebra L ( y, N ) there exists only a finite set of linearlyindependent elements of the sequence (3.5). Hence, there exists a natural M suchthat: W M = λW M − + · · · , (3.10)operators Y , Y , W , . . . , W M − are linearly independent, and three dots stand for alinear combination of the operators Y , Y , W , . . . , W M − . Lemma 2
The operators Y , Y , W are linearly independent.lgebraic Properties of Quasilinear Two-Dimensional Lattices λ W + µ Y + µ Y = 0 . (3.11)The operators Y , Y have the form Y = ∂∂u + · · · , Y = ∂∂u + · · · while W does notcontain terms of the form ∂∂u and ∂∂u , hence the coefficients µ , µ are zero. If, inaddition, λ = 0, then W = 0. We apply the operator ad D x to both sides of the lastequality, then by (3.7) we obtain the equation Y ( a ) Y − Y ( a ) Y = 0which implies: Y ( a ) = α ,u u ,x + γ ,u = 0 and Y ( a ) = α ,u u ,x + γ ,u = 0. By virtueof the independence of the variables u ,x and u ,x , we obtain that α ,u = α ,u = 0. Butthis contradicts the assumption of (1.6) that ∂α ( u n +1 ,u n ,u n − ) ∂u n ± = 0. The proof is complete. Lemma 3
If the expansion of the form (3.10) holds, then α ( u , u , u − ) = P ′ ( u ) P ( u ) + Q ( u − ) + 1 M − Q ′ ( u ) P ( u ) + Q ( u ) − c ( u ) . (3.12)Proof. It is not difficult to show that equation (3.4) for the sequence (3.5) has theform: D x ( λ ) = − a λ − M ( M − Y ( a ) − M Y ( a ) . (3.13)We simplify the relation (3.13) using formulas Y ( a ) = (cid:18) ∂∂u + ( α u ,x + γ ) ∂∂u ,x (cid:19) ( α u ,x + γ ) == (cid:0) α ,u + α (cid:1) u x + γ ,u + α γ , (3.14) Y ( a ) = α ,u u ,x + γ ,u . A simple analysis of the equation (3.13) shows that λ = λ ( u , u ). Therefore, (3.13) isrewritten as λ u u ,x + λ u u ,x = − (cid:18) ( α λ + M ( M − α ,u + α ) (cid:19) u ,x − M α ,u u ,x −− (cid:18) γ λ + M ( M − γ ,u + α γ ) + M γ ,u (cid:19) . Collecting the coefficients in front of the independent variables u ,x , u ,x , we derive anoverdetermined system of differential equations in λ : λ u = − α λ − M ( M − α ,u + α ) , λ u = − M α ,u , (3.15) γ λ + M ( M − γ ,u + α γ ) + M γ ,u = 0 . (3.16)Note that equations (3.15) do not contain function γ and completely coincide with theequations studied in our article [15]. Lemma 3 immediately follows from Lemma 3.2 in[15]. In what follows we use the equation (3.16) to refine the function γ . lgebraic Properties of Quasilinear Two-Dimensional Lattices We construct a test sequence containing the operators Y , Y , Y and their multiplecommutators: Z = Y , Z = Y , Z = Y , Z = [ Y , Y ] , Z = [ Y , Y ] ,Z = [ Y , Z ] , Z = [ Y , Z ] , Z = [ Y , Z ] , Z = [ Y , Z ] . (3.17)Elements of the sequence Z m for m > Z m = [ Y , Z m − ]. Note that this is the simplest test sequence generated by iterations ofthe map Z → [ Y , Z ], which contains the operator [ Y , [ Y , Y ]] = Z . Lemma 4
The operators Z , Z , . . . Z are linearly independent. Proof. Arguing as in the proof of Lemma 1, we can verify that the operators Z , Z , . . . Z are linearly independent. Let us prove the lemma 4 by contradiction.Let’s assume that Z = X j =0 λ j Z j . (3.18)First we derive the formulas by which the operator ad D x acts on the operators Z i . For i = 0 , ,
2, they are immediately obtained from relation[ D x , Y i ] = − a i Y i . Recall that a i = α i u i,x + γ i = α ( u i − , u i , u i +1 ) u i,x + γ ( u i − , u i , u i +1 ). For i = 3 , , D x , Z ] = − ( a + a ) Z + · · · , [ D x , Z ] = − ( a + a ) Z + · · · , [ D x , Z ] = − ( a + a + a ) Z + Y ( a ) Z − Y ( a ) Z + · · · By applying the operator ad D x to both sides of (3.18), we obtain − ( a + a + a )( λ Z + λ Z + · · · ) + Y ( a ) Z − Y ( a ) Z + · · · == λ ,x Z + λ ,x Z − λ ( a + a ) Z − λ ( a + a ) Z + · · · (3.19)Collecting the coefficients for Z in the equality (3.19), we obtain the following equation: λ ,x = − ( α u ,x + γ ) λ − ( α ,u u ,x + γ ,u ) . (3.20)A simple analysis of the equation (3.20) shows that λ = λ ( u , u ). Consequently, λ ,x = λ ,u u ,x + λ ,u u ,x and equation (3.20) reduces to a system of three equations γ λ + γ ,u = 0, λ ,u = − α λ and λ ,u = − α ,u . From these equations it follows that λ = 0. Otherwise, if λ = 0, then α = − (log λ ) u , which implies that ( α ) u − = 0 andthis contradicts the requirement that α ( u , u , u − ) essentially depends on u and u − l ,hence λ = 0. Then from (3.20) we have α ,u = 0, which again leads to a contradiction.Let’s return to the sequence (3.17). For further work, it is necessary to describe theaction of the operator ad D x on all elements of this sequence. It is convenient to separatethe sequence (3.17) into three subsequences { Z m } , { Z m +1 } and { Z m +2 } . lgebraic Properties of Quasilinear Two-Dimensional Lattices Lemma 5
The action of the operator ad D x on the sequence (3.17) is given by thefollowing formulas: [ D x , Z m ] = − ( a + ma ) Z m + (cid:18) m − m Y ( a ) − mY ( a ) (cid:19) Z m − + · · · , [ D x , Z m +1 ] = − ( a + ma ) Z m +1 + (cid:18) m − m Y ( a ) − mY ( a ) (cid:19) Z m − + · · · , [ D x , Z m +2 ] = − ( a + ma + a ) Z m +2 + Y ( a ) Z m +1 + Y ( a ) Z m −− ( m − (cid:16) m Y ( a ) + Y ( a + a ) (cid:17) Z m − + · · · Lemma 5 is easily proved by induction.
Theorem 2
Assume that the operator Z k +2 is represented as a linear combination Z k +2 = λ k Z k +1 + µ k Z k + ν k Z k − + · · · (3.21) of the previous members of the sequence (3.17) and none of the operators Z j +2 for j < k is a linear combination of the operators Z s with s < j + 2 . Then the coefficient ν k satisfies the equation D x ( ν k ) = − a ν k − k ( k − Y ( a ) − ( k − Y ( a + a ) . (3.22) Lemma 6
Assume that all the conditions of Theorem 2 are satisfied. Suppose that theoperator Z k (the operator Z k +1 ) is linearly expressed in terms of the operators Z i , i < k . Then in this expansion the coefficient at Z k − is zero. Proof. Let us prove the assertion by contradiction, assume that in formula Z k = λZ k − + · · · (3.23)the coefficient λ is nonzero. We apply the operator ad D x to both sides of the equation(3.23). As a result, according to Lemma 5, we get: − ( a + ka ) λZ k − + · · · = D x ( λ ) Z k − − λ ( a +( k − a + a ) Z k − + · · · (3.24)Collecting the coefficients at Z k − , we obtain that the coefficient λ should satisfy theequation D x ( λ ) = λ ( a − a ) . According to our assumption above, λ does not vanish and, therefore, D x (log λ ) = a − a . (3.25)Since λ depends on a finite set of dynamical variables, according to the equation (3.25) λ can depend only on u and u . Therefore, from (3.24) we get that(log λ ) u u ,x + (log λ ) u u ,x = α u ,x + γ − α u ,x − γ . The variables u ,x , u ,x are independent, so the last equation is equivalent to thesystem of equations α = − (log λ ) u , alpha = (log λ ) u , γ − γ = 0. Consequently, α = α ( u , u ) depends only on u , u . The latter contradicts the assumption that α lgebraic Properties of Quasilinear Two-Dimensional Lattices u . The contradiction shows that the assumption λ = 0 is false.The lemma is proved.In order to prove Theorem 2, we apply the operator ad D x to both sides of theequality (3.21) and simplify due to the formulas in Lemma 5. Collecting the coefficientsfor Z k − , we obtain the equation (3.22).Let us find the exact values of the coefficients of equation (3.22) Y ( a ) = Y ( α u ,x + γ ) = α ,u u ,x + γ ,u ,Y ( a ) = Y ( α u ,x + γ ) = α ,u u ,x + γ ,u ,Y ( a ) = Y ( α u ,x + γ ) = ( α ,u + α ) u ,x + γ ,u + γ α . and substitute them into (3.22): D x ( ν k ) = − ( α u ,x + γ ) ν k − k ( k − (cid:0) ( α ,u + α ) u ,x + γ ,u + γ α (cid:1) −− ( k − α ,u u ,x + α ,u u ,x + γ ,u + γ ,u ) . (3.26)A simple analysis of the equation (3.26) shows that ν k can depend only on the variables u , u , u . Consequently, D x ( ν k ) = ν k,u u ,x + ν k,u u ,x + ν k,u u ,x . (3.27)Substituting (3.27) in (3.26) and collecting coefficients for independent variables, weobtain a system of equations by the coefficient ν k : ν k,u = − ( k − α ,u , (3.28) ν k,u = − α ν k − k ( k − α ,u + α ) , (3.29) ν k,u = − ( k − α ,u , (3.30)0 = γ ν k + k ( k − γ ,u + γ α ) + ( k − γ ,u + γ ,u ) . (3.31)Substituting the expression for the function α , given by the formula (3.12) into theequation (3.28), we get ν k,u = k − M − P ′ ( u ) Q ′ ( u )( P ( u ) + Q ( u )) . We integrate the last equation with respect to the variable u ν k = − k − M − P ′ ( u ) P ( u ) + Q ( u ) + H ( u , u ) . (3.32)Since ν k,u = H u , the equation (3.30) is rewritten as H u = ( k − P ′ ( u ) Q ′ ( u )( P ( u ) + Q ( u )) . Integrating the latter, we obtain an exact expression for the function HH = − ( k − (cid:18) Q ′ ( u ) P ( u ) + Q ( u ) + A ( u ) (cid:19) , lgebraic Properties of Quasilinear Two-Dimensional Lattices ν k = − ( k − (cid:18) M − P ′ ( u ) P ( u ) + Q ( u ) + Q ′ ( u ) P ( u ) + Q ( u ) + A ( u ) (cid:19) . (3.33)We substitute the found expressions for the functions α and ν k into the equation (3.29) − ( k − M − (cid:18) P ′′ ( u ) P ( u ) + Q ( u ) − P ′ ( u )( P ( u ) + Q ( u )) (cid:19) −− ( k − (cid:18) Q ′′ ( u ) P ( u ) + Q ( u ) − Q ′ ( u )( P ( u ) + Q ( u )) + A ′ ( u ) (cid:19) == ( k − (cid:18) P ′ ( u ) P ( u ) + Q ( u ) + 1 M − Q ′ ( u ) P ( u ) + Q ( u ) − c ( u ) (cid:19) ×× (cid:18) M − P ′ ( u ) P ( u ) + Q ( u ) + Q ′ ( u ) P ( u ) + Q ( u ) + A ( u ) (cid:19) −− k ( k − (cid:18) P ′′ ( u ) P ( u ) + Q ( u ) + 1 M − Q ′′ ( u ) P ( u ) + Q ( u ) −− M − Q ′ ( u )( P ( u ) + Q ( u )) + 1 M − Q ′ ( u ) P ′ ( u )( P ( u ) + Q ( u ))( P ( u ) + Q ( u )) ++ 1( M − Q ′ ( u )( P ( u ) + Q ( u )) − c ′ ( u ) −− c ( u ) (cid:18) P ′ ( u ) P ( u ) + Q ( u ) + 1 M − Q ′ ( u ) P ( u ) + Q ( u ) (cid:19) + c ( u ) (cid:19) . (3.34)Obviously, according to the assumption ∂∂u α ( u , u , u − ) = 0, ∂∂u − l α ( u , u , u − ) = 0the functions P ′ ( u ) and Q ′ ( u ) do not vanish. Consequently, the variables Q ′ ( u )( P ( u ) + Q ( u )) , P ′ ( u )( P ( u ) + Q ( u )) , P ′ ( u ) Q ′ ( u )( P ( u ) + Q ( u ))( P ( u ) + Q ( u ))are independent. Collecting the coefficients of these variables in (3.34), we obtain asystem of two equations (cid:18) − M − (cid:19) (cid:18) − k M − (cid:19) = 0 , M − = kM − . (3.35)The system (3.35) has two solutions: M = 0 , k = − M = 2 , k = 2. Since k mustbe greater than zero, we have M = 2 , k = 2. The last argument completes the proof ofthe theorem 2.Thus, we have proved that M = 2, k = 2. Expansions (3.10) (3.21) take the form W = λW + σY + δY , (3.36) Z = λZ + µZ + νZ + ρZ + κZ + σZ + δZ + ηZ . (3.37)The following is valid Theorem 3
The expansions (3.36), (3.37) take place if and only if the functions α , γ in the equation (1.5) have the form: α ( u n +1 , u n , u n − ) = 1 u n − u n − − u n +1 − u n , (3.38) γ ( u n +1 , u n , u n − ) = r ′ ( u n ) − r ( u n ) α ( u n +1 , u n , u n − ) , (3.39) lgebraic Properties of Quasilinear Two-Dimensional Lattices where r ( u n ) = k u n + k u n + k and the factors k i – are arbitrary constants. Proof. Consider the relation (3.36). Using the relations (3.6), (3.7) and applyingthe Jacobi identity, we get[ D x , W ] = − (2 a + a ) W − Y ( a + 2 a ) W ++(2 Y Y ( a ) − Y Y ( a )) Y − Y Y ( a ) Y . (3.40)It is obvious that only one term in the formula (3.36) contains the operator ofdifferentiation ∂∂u , namely σY , and only one term contains ∂∂u , namely σY .Consequently, σ = 0, δ = 0 and the expansion of (3.36) takes the form W = λW . Applying the operator ad D x to both sides of the last relation, we obtain − (2 a + a ) W − Y ( a + 2 a ) W + (2 Y Y ( a ) − Y Y ( a )) Y − Y Y ( a ) Y == D x ( λ ) W + λ ( − ( a + a ) W + Y ( a ) Y − Y ( a ) Y ) . Collecting the coefficients for the operators W , W , Y , Y , we arrive at the followingsystem: D x ( λ ) = − a λ − Y ( a + 2 a ) , (3.41) − Y Y ( a ) = − λY ( a ) , (3.42)2 Y Y ( a ) − Y Y ( a ) = λY ( a ) . (3.43)Examining the first equation of the obtained system, we observe that λ = λ ( u , u ) andthen simplifying all the equations, we arrive at the following system: λ u = − α λ − ( α ,u + α ) , (3.44) λ u = − α ,u , (3.45) α ,u u = λα ,u , α ,u u = λα ,u . (3.46) γ λ + γ ,u + γ α + 2 γ ,u = 0 , (3.47) γ ,u u = λγ ,u , (3.48) γ ,u u + γ α ,u − γ ,u α = λγ ,u . (3.49)We note at once that the equations (3.44)-(3.46) will be used to refine the functions α and λ , and the equations (3.47)-(3.49) to specify the function γ , substituting the alreadyfound expression for α .Next, we turn to the decomposition (3.37). Putting k = 2 in the formulas ofLemma 5, we obtain[ D x , Z ] = − ( α u ,x + 2 α u ,x ) Z + · · · , (3.50)[ D x , Z ] = − ( α u ,x + 2 α u ,x ) Z − ( Y ( α u ,x ) + 2 Y ( α u ,x )) Z + · · · , (3.51)[ D x , Z ] = − ( α u ,x + 2 α u ,x + α u ,x ) Z + Y ( α u ,x ) Z + Y ( α u ,x ) Z −− ( Y ( α u ,x ) + Y ( α u ,x + α u ,x )) Z + · · · . (3.52) lgebraic Properties of Quasilinear Two-Dimensional Lattices D x to both parts of the relation (3.37) and simplify theresulting equation using (3.50), (3.51), (3.52). Comparison of coefficients at Z and Z gives λ = 0 and µ = 0. Thus, the formula (3.37) is simplified: Z = νZ + ρZ + κZ + σZ + δZ + ηZ . (3.53)The following commutation relations hold:[ D x , Z ] = − ( a + 2 a + a ) Z + Y ( a ) Z − Y ( a ) Z − Y ( a + a + a ) Z ++ Y Y ( a ) Z − Y Y ( a ) Z + ( Y Y Y ( a ) + Z ( a )) Z , (3.54)[ D x , Z ] = − ( a + a + a ) Z + Y ( a ) Z − Y ( a ) Z + Y Y ( a ) Z . (3.55)We apply ad D x to (3.53), then simplify according to (3.54), (3.55), (3.53) and collectthe coefficients for Z − ( a + 2 a + a ) ν − Y ( a + a + a ) = D x ( ν ) − ( a + a + a ) ν or the same D x ( ν ) = − a ν − Y ( a + a + a ) . (3.56)From the equation (3.56) it follows that ν depends on three variables ν = ν ( u, u , u ).Thus, the equation (3.56) reduces to a system of equations: ν u = − α ,u , (3.57) ν u = − α ν − α ,u − α , (3.58) ν u = − α ,u , (3.59) γ ν + γ ,u + γ α + γ ,u + γ ,u = 0 . (3.60)So, as a result of the investigation of the relations (3.36), (3.37), we come to the equations(3.44)-(3.46) and (3.57)-(3.59), which exactly coincide with the corresponding systemsof equations from the work [15] and, therefore, we get that α ( u n +1 , u n , u n − ) = 1 u n − u n − − u n +1 − u n . Using the remaining equations (3.47)-(3.49) and (3.60) we find γ : γ ( u n +1 , u n , u n − ) = r ′ ( u n ) − r ( u n ) α ( u n +1 , u n , u n − ) . It is not difficult to show that the relations (3.36), (3.37) take the form: W = λW , λ = 2 u − u ,Z = νZ , ν = − u − u + u ( u − u )( u − u ) . Similarly, we have β ( u n +1 , u n , u n − ) = ˜ r ′ ( u n ) − ˜ r ( u n ) α ( u n +1 , u n , u n − ) , (3.61)where ˜ r ( u n ) = ˜ k u n + ˜ k u n + ˜ k , and the coefficients ˜ k i – are arbitrary constants.The next step of our investigation is to refine the function δ . To do this, we builda new sequence on a set of multiple commutators. lgebraic Properties of Quasilinear Two-Dimensional Lattices
4. Specification of the function δ Recall that since the right-hand side f i of the system (1.5) has a special form, theoperator Y can be represented as follows (see (2.5)): Y = N X i =0 u i,y Y i + R, Here the operator R is defined by the formula (2.6). Consider the following sequence ofthe operators in the characteristic algebra L ( y, N ): Y − , Y , Y , Y , − = [ Y , Y − ] , Y , = [ Y , Y ] , (4.1) R = [ Y , R ] , R = [ Y , R ] , R = [ Y , R ] , . . . , R k +1 = [ Y , R k ] . The following commutation relations hold:[ D x , Y − ] = − a − Y − , [ D x , Y ] = − a Y , [ D x , Y ] = − a Y , (4.2)[ D x , Y , ] = − ( a + a ) Y , − Y ( a ) Y + Y ( a ) Y , (4.3)[ D x , Y , − ] = − ( a − + a ) Y , − − Y ( a − ) Y − + Y − ( a ) Y , (4.4)[ D x , R ] = − X i h i Y i , (4.5)where a i = α i u i,x + γ i , h i = β i u i,x + δ i . Using the Jakobi identity and formulas (4.2)–(4.5)we derive the formulas:[ D x , R ] = [ D x , [ Y , R ]] = − [ Y , [ R, D x ]] − [ R, [ D x , Y ]] == − a R + h Y , − h − Y , − −− Y ( h ) Y − Y ( h − ) Y − + ( R ( a ) − Y ( h )) Y , (4.6)[ D x , R ] = − a R − Y ( a ) R + · · · , (4.7)[ D x , R ] = − a R − Y ( a ) R − Y ( a ) R + · · · , (4.8)[ D x , R ] = − a R − Y ( a ) R − Y ( a ) R − Y ( a ) R + · · · , (4.9)where three dots stand for a linear combination of the operators Y , , Y , − , Y , Y , Y − .It can be proved by induction that[ D x , R n ] = p n R n + q n R n − + · · · , (4.10)where p n = − ( n + 1) a , q n = − n + n Y ( a ) , (4.11)and three dots stand for a linear combination of the operators R k , k < n − Y , , Y , − , Y , Y , Y − .Now we consider two different cases: i ) The operator R is linearly expressed in terms of the opeartors (4.1). ii ) The operator R is not linearly expressed in terms of the operators (4.1).Let us focus on the case i ). It follows from the formula (4.6) that this linearexpansion must have the form R = λR + µY , + ˜ µY , − + νY + ηY + ǫY − . (4.12) lgebraic Properties of Quasilinear Two-Dimensional Lattices D x to both sides of (4.12), we obtain − a ( λR + µY , + ˜ µY , − + · · · ) + h Y , − h − Y , − + · · · == D x ( λ ) R + D x ( µ ) Y , + µ ( − ( a + a ) Y , + · · · ) ++ D x (˜ µ ) Y , − + ˜ µ ( − ( a − + a ) Y , − + · · · ) . (4.13)Three dots stand for a linear combination of the operators Y − , Y , Y . Collectingcoefficients for the independent operators R , Y , , Y , − , we obtain the system ofdifferential equations for the coefficients λ , µ , ˜ µD x ( λ ) = − a λ, (4.14) D x ( µ ) = a µ + h , D x (˜ µ ) = a − ˜ µ − h − . (4.15)The equation (4.14) has the form: D x ( λ ) = − ( α u ,x + γ ) λ . It is easy to see that λ = λ ( u ) and hence λ ′ ( u ) = − α λ ( u ), 0 = γ λ . If λ = 0 then α = − (log λ ( u )) ′ .But this contradicts the assumption (1.6) requesting that ∂α ( u ,u ,u − ) ∂u ± = 0. Hence wehave λ = 0. Now consider the equations (4.15): D x ( µ ) = ( α u ,x + γ ) µ + β u ,x + δ , (4.16) D x (˜ µ ) = ( α − u − ,x + γ − )˜ µ − β − u − ,x − δ − . (4.17)From (4.16) we obtain that µ depends only on u and from (4.17) we obtain that ˜ µ depends only on u − . Now the equations (4.16) and (4.17) are reduced to the followingsystem of equations: µ ′ ( u ) = α µ ( u ) + β , γ µ ( u ) + δ , (4.18)˜ µ ( u − ) = α − ˜ µ ( u − ) − β − , γ − ˜ µ ( u − ) − δ − . (4.19)By shifting the argument n backwards and forwards by one in the equation (4.18) and,respectively, in (4.19) we obtain. µ ′ ( u ) = α µ ( u ) + β , γ µ ( u ) + δ , (4.20)˜ µ ( u ) = α ˜ µ ( u ) − β , γ ˜ µ ( u ) − δ . (4.21)Now we exlude µ and ˜ µ from these equations and arrive at the differential equation forthe function δ : δ ,u = (cid:18) γ ,u γ + α (cid:19) δ − β γ . (4.22)Equation (4.22) is easily solved δ ( u − , u , u ) = 14 1( u − u − )( u − u ) ×× (cid:16) k ( u u − u u − u + u u − ) + 2 k ( u − u − u ) + 2 k ( − u + 2 u − u − ) (cid:17) ×× (cid:16) ˜ k ( u u − u − u + u − u ) + 2˜ k u + 2˜ k + 4 F ( u − , u )( u − u − )( u − u ) (cid:17) . (4.23)Here k , k , k and ˜ k , ˜ k , ˜ k are constants, which appear in the description of functions(3.39), (3.61) and F ( u − , u ) is a function to be found. lgebraic Properties of Quasilinear Two-Dimensional Lattices F ( u − , u ) = ˜ k and µ ( u ) = 12 ˜ k u + ˜ k u + ˜ k From the second equation of (4.21) we get˜ µ ( u ) = −
12 ˜ k u − ˜ k u − ˜ k . Further, we collect the coefficients for Y , Y − in the equality (4.14). Substitutingthe functions α , β , γ , δ , µ , ˜ µ found above into the equations obtained we get identitiesthat do not give any additional condition on the unknown functions. Let us collect thecoefficients for Y in (4.14) and find: D x ( η ) = R ( a ) − Y ( h ) + µY ( a ) − ˜ µY − ( a ) . Calculating each term and simplifying the last equation, we obtain D x ( η ) = ( − β ,u + µα ,u − ˜ µα ,u − ) u ,x + δ α − δ ,u − γ β + µγ ,u − ˜ µγ ,u − . A simple analysis of the last equation shows that η can depend only on u . Therefore,the last equation reduces to a system of two equations: η ′ ( u ) = − β ,u + µα ,u − ˜ µα ,u − , (4.24)0 = δ α − δ ,u − γ β + µγ ,u − ˜ µγ ,u − . (4.25)By direct calculation we obtain that the right hand side of (4.24) is identically equal tozero.Investigating the equation (4.25) we derive some additional relations between theconstants k i ˜ k i : k = ˜ k ˜ k k , k = ˜ k ˜ k k . (4.26)Thus, it is proved that if the decomposition (4.12) takes place then it should be of theform R = µY , + ˜ µY , − . (4.27)Herewith we completely determine the desired coefficients of the quasilinear chain(1.5) u n,xy = α n u n,x u n,y + β n u n,x + γ n u n,y + δ n . (4.28)Summarizing the reasonings above we present the explicit expressions for thesecoefficients α n = α ( u n +1 , u n , u n − ) = 1 u n − u n − − u n +1 − u n ,β n = β ( u n +1 , u n , u n − ) = r ′ ( u n ) − r ( u n ) α ( u n +1 , u n , u n − ) ,γ n = γ ( u n +1 , u n , u n − ) = ε ( r ′ ( u n ) − r ( u n ) α ( u n +1 , u n , u n − )) ,δ n = δ ( u n +1 , u n , u n − ) = − εr ( u n )( r ′ ( u n ) − r ( u n ) α ( u n +1 , u n , u n − )) , lgebraic Properties of Quasilinear Two-Dimensional Lattices r ( u n ) = c u n + c u n + c is a polynomial of the degree not higher then two witharbitrary coefficients and c i = ˜ k i , ε = k / ˜ k . The boundary conditions reducing thechain to an integrable system of hyperbolic equations are given in the form u − = λ, u N +1 = λ (4.29)where λ is a root of the polynomial r ( λ ) i.e. r ( λ ) = 0. In the degenerate case when r ( u n ) = c the boundary conditions are of the form u − = c ( εx + y ) + c , u N +1 = c ( εx + y ) + c , (4.30)where c , c are arbitrary constants.Let us investigate the case ii ). Assume that some element R n , n > R n = λR n − + · · · , (4.31)but elements R k , k < n are not expressed linearly in terms of the previous elements R j , j < k and Y , , Y , − , Y , Y , Y − . Let us apply the operator ad D x to both sides of (4.31)and obtain p n ( λR n − + · · · ) + q n R n − + · · · = D x ( λ ) R n − + λ ( p n − R n − + · · · ) . We collect the coefficients for the operator R n − in the resulting equality and find: D x ( λ ) = λ ( p n − p n − ) + q n . We substitute explicit expressions for p n , p n − , q n into the last equation and get D x ( λ ) = − a λ − n + n Y ( a ) . We substitute the explicit expression for a and evaluate Y ( a ). So we obtain D x ( λ ) = − ( α u ,x + γ ) λ − n + n (cid:0) ( α ,u + α ) u ,x + γ ,u + γ α (cid:1) (4.32)It follows from the last equality that λ depends only on u . Then the equation reducesto a system of two equations λ ′ ( u ) = − α λ − n + n α ,u + α ) , (4.33) γ λ + n + n γ ,u + γ α ) = 0 . (4.34)Rewrite the equation (4.33) as follows λ ′ ( u ) = − λ ( u )(2 u − u − u − ) − ( n + n )( u − u − )( u − u ) (4.35)or λ ′ ( u )( u − u u − u − u + u u − ) = − λ ( u )(2 u − u − u − ) − ( n + n ) . (4.36)Since the variables u − , u , u are independent then the last equation impliesimmediately that λ = 0 and n + n = 0. Thus we have n = 0 or n = −
1. Bothsolutions contradict the assumption n >
0. Therefore the case ii ) is never realized. lgebraic Properties of Quasilinear Two-Dimensional Lattices c = c = 0, then by the shift transformation u → u − c ( εx + y ) the chain(4.28) reduces to the known Ferapontov-Shabat-Yamilov chain (see [27, 28]) u n,xy = α n u n,x u n,y , (4.37)2) If c = 0, c = 0, then by shifting u → u − c c and stretching x → xεc , y → yc weobtain the chain u n,xy = α n ( u n,x u n,y − u n ( u n,x + u n,y ) + u n ) + u n,x + u n,y − u n , (4.38)3) For c = 0 by the shift transformation u → u − c c and by the stretching x → εc x , y → c y chain (4.28) can be reduced to the form u n,xy = α n ( u n,x u n,y − s n ( u n,x + u n,y ) + s n ) + s ′ n ( u n,x + u n,y − s n ) , (4.39)where s n = u n + C and C = c c − (cid:16) c c (cid:17) – is an arbitrary constant.Thus we have proved that any chain integrble in the sense of our Definition 1 isof the form (4.28). In order to complete the proof of Theorem 1 we have to verify theconverse statement. It is done in the following theorem. Theorem 4
The chain (4.28), found as a result of the classification, is integrable inthe sense of Definition 1 formulated in the Introduction.
We introduce special notations for multiple commutators of the operators { Y i } Y i k ,...,i = [ Y i k , Y i k − ,...,i ] . (4.40)The structure of the Lie algebra generated by the operators { Y i } can be studied by themethod developed in our previous paper [15]. One can prove that any element of thisalgebra can be represented as a linear combination of the following operators Y i , Y i +1 ,i , Y i +2 ,i +1 ,i , . . . . (4.41)It follows from the formula (2.5) that the algebra L ( y, N ) corresponding to the system(2.1) is an extension of this algebra, obtained by adding one more generator, namely,the operator R .Recall that in the paper [15] the particular case of a chain (4.28) was studied indetail. Namely, the following theorem was proved: Theorem 5
The chain u n,xy = (cid:18) u n − u n − − u n +1 − u n (cid:19) u n,x u n,y (4.42) is integrable by Definition 1, formulated in the Introduction. Recall briefly the scheme of the proof of the Theorem 5. The basis { Y i } Ni =0 , { Y i +1 ,i } N − i =0 , { Y i +2 ,i +1 ,i } N − i =0 , . . . , Y N,N − ,..., . (4.43) lgebraic Properties of Quasilinear Two-Dimensional Lattices Y , ..., Y N corresponding to the chain (4.42).In order to prove that there is a basis (4.43) on the set of multiple commutators ofthe operators Y , ..., Y N corresponding to the chain (4.28) we can repeat the proof of theTheorem 5.2 from the paper [15] (see Appendix), putting a i = α i u i,x + γ i . These proofis cumbersome, so we do not give it here.In order to prove the main Theorem 4 we consider the algebra Lie L ( y, N ) generatedby the operators Y , ..., Y N , R and we prove that the finite basis exists in this algebra R, { Y i } Ni =0 , { Y i +1 ,i } N − i =0 , { Y i +2 ,i +1 ,i } N − i =0 , . . . , Y N,N − ,..., . (4.44)So it remains to verify that any multiple commutator of the operator R with theoperators (4.43) is linearly expressed in terms of the operators from the set (4.44).Let us prove the Theorem 4.Proof. Here we consider truncated chains, i.e. finite systems of the hyperbolicequations (2.1), obtained by imposing cut-off conditions to the initial chain. Note thatcommutation relations near the cut-off points change in the transition from an infinitechain to a truncated one.We prove the Theorem 4 by induction. Let us justify the induction base. The firststep of the proof requires the following formulas: (cid:2) D x , ¯ R (cid:3) = − a ¯ R + h Y , − Y ( h ) Y + ( R ( a ) − Y ( h )) Y , (4.45) (cid:2) D x , ¯ R N (cid:3) = − a N ¯ R N − h N − [ Y N , Y N − ] −− Y N ( h N − ) Y N − + ( R ( a N ) − Y N ( h N )) Y N , (4.46) (cid:2) D x , ¯ R k (cid:3) = − a k ¯ R k − h k − [ Y k , Y k − ] + h k +1 [ Y k +1 , Y k ] −− Y k ( h k − ) Y k − + ( R ( a k ) − Y k ( h k )) Y k − Y k ( h k +1 ) Y k +1 . (4.47)Here ¯ R j = [ Y j , R ], j = 0 , . . . , N .At first we study the end points k = 0 and k = N . Let us show that the followingequality holds ¯ R = λ (0) R + µ (0) Y , + ν (0) Y + η (0) Y . (4.48)We apply the operator ad D x to both sides of the equality (4.48) and simplify using (4.2),(4.3), (4.5), (4.45), as a result we obtain − a ( λ (0) R + µ (0) Y , + · · · ) + h Y , + · · · == D x ( λ (0) ) R + D x ( µ (0) ) Y , + µ (0) ( − ( a + a ) Y , + · · · ) . (4.49)Here three dots stand for a linear combination of the operators Y , Y . Collecting thecoefficients for the operators R and Y , in (4.49), we obtain a system of the equations D x ( λ (0) ) = − a λ (0) , (4.50) D x ( µ (0) ) = a µ (0) + h . (4.51)The equation (4.50) coincides with equation (4.14) for i = 0, hence λ (0) = 0. Theequation (4.51) coincides with the first equation (4.15) for i = 0, hence µ (0) = µ . It is lgebraic Properties of Quasilinear Two-Dimensional Lattices ν (0) = η (0) = 0. Thus we have proved that the decomposition (4.48)has the form ¯ R = µ (0) Y , . (4.52)Let us show that the following equality holds¯ R N = λ ( N ) R + ˜ µ ( N ) Y N,N − + η ( N ) Y N + ǫ ( N ) Y N − . (4.53)We apply ad D x to both sides of the relation (4.53): − a N ( λ ( N ) R + ˜ µ ( N ) Y N,N − + · · · ) − h N − Y N,N − + · · · == D x ( λ ( N ) ) R + D x (˜ µ ( N ) ) Y N,N − + ˜ µ ( N ) ( − ( a N + a N − ) Y N,N − + · · · ) . (4.54)Here three dots stand for a linear combination of the operators Y N , Y N − . Collectingthe coefficients for R and Y N,N − , we get the system: D x ( λ ( N ) ) = − a N λ ( N ) , (4.55) D x (˜ µ ( N ) ) = a N − ˜ µ ( N ) − h N − . (4.56)The equation (4.55) coincides with equation (4.14) for i = N , hence λ ( N ) = 0. Theequation (4.56) coincides with the second equation (4.15) for i = N , hence we have˜ µ ( N ) = D Nn ˜ µ ( u − ) = ˜ µ ( u N − ). It is easy to show that η ( N ) = ǫ ( N ) = 0. So we haveproved that the decomposition (4.53) has the form:¯ R N = ˜ µ ( N ) Y N,N − . (4.57)Now we concentrate on the inner points by taking k from the set 0 < k < N . Letus show that the following equality holds¯ R k = λ ( k ) R + µ ( k ) Y k +1 ,k + ˜ µ ( k ) Y k,k − + ν ( k ) Y k +1 + η ( k ) Y k + ǫ ( k ) Y k − . (4.58)We apply the operator ad D x to both sides of the relation (4.58): − a k ( λ ( k ) R + µ ( k ) Y k +1 ,k + ˜ µ ( k ) Y k,k − + · · · ) − h k − Y k,k − + h k +1 Y k +1 ,k + · · · == D x ( λ ( k ) ) R + D x ( µ ( k ) ) Y k +1 ,k + D x (˜ µ ( k ) ) Y k,k − ++ µ ( k ) ( − ( a k +1 + a k ) Y k +1 ,k + · · · ) + ˜ µ ( k ) ( − ( a k + a k − ) Y k,k − + · · · ) . (4.59)Here three dots stand for a linear combination of the operators Y , Y , ..., Y N − , Y N .Collecting the coefficients for R , Y k +1 ,k , Y k,k − in (4.59), we obtain the system D x ( λ ( k ) ) = − a k λ ( k ) , (4.60) D x ( µ ( k ) ) = a k +1 µ ( k ) + h k +1 , (4.61) D x (˜ µ ( k ) ) = a k − ˜ µ ( k ) − h k − . (4.62)The equation (4.60) coincides with (4.14) if i = k . That is why we obtain that λ ( k ) = 0.The equation (4.61) coincides with the first equation (4.15) if i = k , and equation (4.62)coincides with the second equation (4.15) if i = k . Hence, µ ( k ) = D kn ( µ ( u )) = µ ( u k +1 ),˜ µ ( k ) = D kn (˜ µ ( u − )) = ˜ µ ( u k − ). It is easy to show that ν ( k ) = η ( k ) = ǫ ( k ) = 0. Thus, wehave proved that the decomposition (4.58) has the form:¯ R k = µ ( k ) Y k +1 ,k + ˜ µ ( k ) Y k,k − . (4.63) lgebraic Properties of Quasilinear Two-Dimensional Lattices Y i +1 ,i , R ] for some i , 0 ≤ i ≤ N −
1. Using theJakobi identity, we obtain[ Y i +1 ,i , R ] = − [ R, Y i +1 ,i ] = − [ R, [ Y i +1 , Y i ]] == [ Y i +1 , [ Y i , R ]] + [ Y i , [ R, Y i +1 ]] == (cid:2) Y i +1 , µ ( i ) Y i +1 ,i + ˜ µ ( i ) Y i,i − (cid:3) − (cid:2) Y i , µ ( i +1) Y i +2 ,i +1 + ˜ µ ( i +1) Y i +1 ,i (cid:3) == Λ ( i ) Y i +2 ,i +1 ,i + M ( i ) Y i +1 ,i,i − + κ ( i ) Y i +2 ,i +1 + η ( i ) Y i +1 ,i + ζ ( i ) Y i,i − , where Λ ( i ) , M ( i ) , κ ( i ) , η ( i ) , ζ ( i ) – some functions that depend on dynamic variables.Herewith ζ (0) = 0, M (0) = 0, Λ N − = 0, κ N − = 0.Now let us justify the inductive transition. Assume that for a given M , 0 ≤ k M,M − ,...,k,k − ++ νY M,M − ,...,k + εY M +1 ,M,M − ,..,k +1 + ηY M − ,...,k,k − ++ ζ Y M − ,M − ,...,k + θY M,M − ,...,k +1 + ξY M − ,M − ,...,k − + · · · ++ · · · + κY M +1 ,M + ϕY M,M − + · · · + χY k,k − . (4.64)Let us show that a similar representation holds for M + 1. Using the Jakobi identity,we obtain that the following decomposition holds:[ Y M +1 ,M,M − ,...,k , R ] = − [ R, [ Y M +1 , Y M,M − ,...,k ]] == [ Y M +1 , [ Y M,M − ,...,k , R ]] + [ Y M,M − ,..,k , [ R, Y M +1 ]] == [ Y M +1 , [ Y M,M − ,...,k , R ]] − [ Y M,M − ,..,k , R M +1 ] . We substitute the decomposition (4.64) and a proper one of the equations (4.63), (4.52)or (4.57) (it depends on the concrete value of M : M = 0, M = N or 0 < M < N ) intothe last formula. Then we expand the commutators using the linearity property. Thelatter completes the proof of Theorem 4. Conclusions In this paper the problem of the integrable classification of two-dimensional chains ofthe type (1.1) is studied. For chains of a special type (1.5), (1.6) a complete descriptionof the integrable cases is obtained. By integrability of the chain we mean here theexistence of reductions in the form of arbitrarily high order systems of hyperbolic typeequations that are Darboux integrable. 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