Lax pair for one novel two-dimensional lattice
aa r X i v : . [ n li n . S I] F e b Symmetry, Integrability and Geometry: Methods and Applications SIGMA * (201*), ***, 12 pages Lax pair for one novel two-dimensional lattice
Mariya N. KUZNETSOVAInstitute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences, 112Chernyshevsky Street, Ufa 450008, Russia
E-mail: [email protected]
Received ???, in final form ????; Published online ????http://dx.doi.org/10.3842/SIGMA.201*.***
Abstract.
In our recent papers [1, 2] the algorithm for classification of integrable equationswith three independent variables was proposed. This method is based on the requirementof the existence of an infinite set of Darboux integrable reductions and on the notion ofthe characteristic Lie-Rinehart algebras. The method was applied for the classification ofintegrable cases of different subclasses of equations u n,xy = f ( u n +1 , u n , u n − , u n,x , u n,y ) , −∞ < n < ∞ of special forms [3, 4, 5, 6]. Under this approach the novel integrable chain u n,xy = α n ( u n,x − u n − u n,y − u n −
1) + 2 u n ( u n,x + u n,y − u n − α n = u n − u n − − u n +1 − u n , the sought function u n = u n ( x, y ) dependson the real x, y and the integer n . In present paper we construct Lax pair for this chain. Toconstruct the Lax pair, we use the scheme suggested in paper [7]. We verified that the foundLax pair is not “fake” by studying the periodic reduction of the chain and the Lax pair.For the system of the hyperbolic type equations obtained from the chain we constructedgeneralized symmetry of the second order which has unusual structure. The matter is thatthe system admits an integral and the symmetry also depends on this integral. We made asimilar analysis for the Ferapontov-Shabat-Yamilov chain u n,xy = α n u n,x u n,y as well. Key words:
Lax pair, two-dimensional lattice, integrable reduction, characteristic algebra,Lie-Rinehart algebra, Darboux integrable system, higher symmetry, x -integral A number of our recent publications [1, 2, 3, 4, 5, 6] are addressed the problem of integrableclassification of two-dimensional lattices u n,xy = f ( u n +1 , u n , u n − , u n,x , u n,y ) , −∞ < n < ∞ , (1.1)where the sought function u n = u n ( x, y ) depends on the real x, y and the integer n . In thesepapers we proposed the method for seeking and classifying integrable equations with threeindependent variables based on the requirement of the existence of a set of Darboux integrablereductions and on the notion of the characteristic Lie-Rinehart algebras. The method wasapplied to different subclasses of equations (1.1) of special forms.Within this approach we use the following Definition 1.
A lattice of the form (1.1) is called integrable if there exist locally analyticfunctions ϕ and ψ of two variables such that for any choice of integers N , N the hyperbolic M.N. Kuznetsovatype system u N ,xy = ϕ ( u N +1 , u N ) ,u n,xy = f ( u n +1 , u n , u n − , u n,x , u n,y ) , N < n < N , (1.2) u N ,xy = ψ ( u N , u N − ) , obtained from lattice (1.1) by imposing cut-off conditions at n = N and n = N , is integrablein the sense of Darboux.Let us recall what Darboux integrability means. Definition 2.
A function I = I ( x, ¯ u, ¯ u x , ¯ u xx , ... ) is called an y -integral if it satisfies the equation D y I = 0 for every solution of system (1.2). A function J = J ( y, ¯ u, ¯ u y , ¯ u yy , ... ) is called a x -integral if it satisfies the equation D x J = 0. Integrals of the form I = I ( x ) and J = J ( y ) arecalled trivial.Here ¯ u is a vector ¯ u = ( u N , u N +1 , . . . , u N ), ¯ u x is its derivative and so on. The operators D y and D x are operators of the total derivative with respect to the variable y or x , correspondingly,by virtue of system (1.2). Definition 3.
A system (1.2) is called Darboux integrable if it possesses N − N +1 functionallyindependent nontrivial integrals in both characteristic directions x and y .Darboux integrable systems are amenable to study by the Lie-Rinehart algebras. Let I = I ( x, ¯ u, ¯ u x , ¯ u xx , ... ) be a nontrivial y -integral for the system (1.2). Then I must satisfy thefollowing system: Y I = 0 , X i I = 0 , where X i = ∂∂u i,y , Y = N X i = N (cid:18) u i,y ∂∂u i + f i ∂∂u i,x + D x ( f i ) ∂∂u i,xx + · · · (cid:19) and f i = f ( u i +1 , u i , u i − , u i,x , u i,y ). The first equation follows from the fact that the operator D y acts on functions I = I ( x, ¯ u, ¯ u x , ¯ u xx , ... ) by the rule D y I = Y I , the second one arises because I doesn’t depend on variables u i,y .Let us consider the Lie algebra L y generated by the operators Y , X i over the ring A of locallyanalytic functions of the dynamical variables ¯ u y , ¯ u, ¯ u x , ¯ u xx , . . . . To the standard multiplicationoperation [ Z, W ] = ZW − W Z we add two conditions: [
Z, aW ] = Z ( a ) W + a [ Z, W ] and ( aZ ) b = aZ ( b ) valid for any Z, W ∈ L y and a, b ∈ A . These equalities means that for any Z ∈ L y andany a ∈ A , the element aZ ∈ L y . In this case the algebra L y is called the Lie-Rinehart algebra[9], [10].If there exists a finite basis Z , Z , . . . , Z k ∈ L y such that an arbitrary element Z ∈ L y is repre-sented as a linear combination Z = a Z + a Z + · · · + a k Z k , where coefficients a , a , . . . , a k ∈ A ;and if the equality Z = 0 implies that a = a = . . . = a k = 0, then algebra L y is of a finitedimension.The integrability criterion of the hyperbolic type system in the sense of Darboux is formulatedas follows [11, 12]: Theorem 1.
System (1.2) admits a complete set of the y -integrals (a complete set of the x -integrals) if and only if its characteristic algebra L y (respectively, characteristic algebra L x ) isof finite dimension. ax pair for one novel two-dimensional lattice 3 Corollary 1.
System (1.2) is integrable in the sense of Darboux if both characteristic algebras L x and L y are of finite dimension. The above statements play a key role in our classification works. Within the scope of thispaper we need one of our results: paper [4] provides a complete list of integrable two-dimensionallattices 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 − ) , (1.3)with the coefficient α satisfying the conditions ∂α ( u n +1 ,u n ,u n − ) ∂u n ± = 0. This list consists of threeequations: Theorem 2.
Integrable equation of the form (1.3) can be reduced by a point transformation toone of the following forms: u n,xy = α n u n,x u n,y , α n = 1 u n − u n − − u n +1 − u n = u n +1 − u n + u n − ( u n +1 − u n )( u n − u n − ) , (1.4) u n,xy = α n ( u n,x − u n )( u n,y − u n ) + u n,x + u n,y − u n , (1.5) u n,xy = α n ( u n,x − u n − u n,y − u n −
1) + 2 u n ( u n,x + u n,y − u n − . (1.6)Equation (1.4) was found before in papers [13], [14] by Ferapontov and Shabat and Yamilov.Equations (1.5) and (1.6) appeared in [4] as a result of the classification procedure. Equation(1.5) is reduced to (1.4) by the point transformation u n = e x + y v n .The aim of the paper is to find Lax pair for novel chain (1.6), to explain the method of findingLax pairs and to prove that periodic closings of the chain possesses higher symmetries.The Lax pair for equation (1.4) ψ n,x = u n,x u n +1 − u n ( ψ n +1 − ψ n ) , ψ n,y = u n,y u n − u n − ( ψ n − ψ n − ) (1.7)was found by E.V. Ferapontov. To construct Lax pair for chain (1.6), we use the schemesuggested in paper [7]. Let us describe the procedure in detailed. First of all, we representlattice (1.6) in the equivalent following form: u xy = ( u x − u − u y − u − △ z ¯ z u △ z u △ ¯ z u + 2 u ( u x + u y − u − . (1.8)Here △ z = T z − ǫ , △ ¯ z = − T ¯ z ǫ are the forward/backward discrete derivatives and △ z ¯ z = T z + T ¯ z − ǫ is the symmetrised second-order discrete derivative; the operators T z , T ¯ z are the forward andbackward ǫ -shifts operators in the variable z .The method consists of three steps:(1) First we construct the dispersionless limit of the equation (obtained as ǫ → u n +2 = u n to infinite chains (1.4), (1.6) and obtain finite systems. Lax pairs and higher sym-metries of the second order are constructed for obtained finite systems. Conclusion contains adiscussion of the results. M.N. Kuznetsova (1.6) The main result of this section is as follows:
Theorem 3.
Equation (1.6) possesses the Lax pair ψ n,x = u n,x − u n − u n +1 − u n ( ψ n +1 − ψ n ) + u n ψ n ,ψ n,y = u n,y − u n − u n − u n − ( ψ n − ψ n − ) + u n ψ n . Proof .
The dispersionless limit of the equation (1.8) coincides with equation: u xy = ( u x − u − u y − u − u zz u z + 2 u ( u x + u y − u − . (2.1)There exists a direct method for finding Lax pairs for equations of this form. Lax pair is soughtin the following form: S x = F ( u, u x , u y , u z , S z ) , (2.2) S y = G ( u, u x , u y , u z , S z ) . (2.3)The compatibility condition S xy = S yx of system (2.2), (2.3) by virtue of equation (2.1) leadsto the overdetermined equation F u y u yy u z − G u x u xx u z − ( G S z F u x − G u x F S z + G u z ) u zx u z − (cid:0) G u y F S z − G S z F u x − F u z (cid:1) u zy u z ++ u zz (cid:0) ( u − u y + 1)( u − u x + 1) (cid:0) F u x − G u y (cid:1) − u z ( G S z F u z − G u z F S z ) (cid:1) −− u z (cid:0) u (1 + u − u x − u y ) (cid:0) F u x − G u y (cid:1) + u z ( G S z F u − G u F S z ) + u x G u − u y F u (cid:1) = 0 . Because of the fact that variables u, u x , u y , u z , u xx , u yy , u zx , u zy , u zz are independent, this equa-tion splits down into the overdetermined system of equations: F u y = 0 , G u x = 0 , (2.4) G S z F u x − G u x F S z + G u z = 0 , (2.5) G u y F S z − G S z F u x − F u z = 0 , (2.6)( u − u y + 1)( u − u x + 1) (cid:0) F u x − G u y (cid:1) − u z ( G S z F u z − G u z F S z ) = 0 , (2.7)2 u (1 + u − u x − u y ) (cid:0) F u x − G u y (cid:1) + u z ( G S z F u − G u F S z ) + u x G u − u y F u = 0 . (2.8)Equations (2.4) mean that F = F ( u, u x , u z , S z ) and G = G ( u, u y , u z , S z ). Substituting F and G into (2.5), (2.6), we arrive at the equations: G u z + G S z F u x = 0 , F u z + G u y F S z = 0 . (2.9)We differentiate the first equation (2.9) by u x , the second equation (2.9) – by u y , and obtainthat G S z F u x u x = 0, F S z G u y u y = 0. Obviously that the functions F and G take the followingforms: F ( u, u x , u z , S z ) = F ( u, u z , S z ) u x + F ( u, u z , S z ) ,G ( u, u y , u z , S z ) = F ( u, u z , S z ) u y + F ( u, u z , S z ) . Then we rewrite (2.9) and (2.7), (2.8) using the last formulas. Because of the fact that thevariables u, u x , u y , u z are independent, obtained equations split down one more time. Thus wearrive at the system for unknown functions F i ( u, u z , S z ), i = 2 , , , F F ,S z + F ,u z = 0 , F F ,S z + F ,u z = 0 (2.10)ax pair for one novel two-dimensional lattice 5 F − F + u z ( F ,S z F ,u z − F ,S z F ,u z ) = 0 , (2.11) F ,u − F ,u + u z ( F ,u F ,S z − F ,u F ,S z ) = 0 , (2.12) F F ,S z + F ,u z = 0 , (2.13)(1 + u )( F − F ) + u z ( F ,S z F ,u z − F ,S z F ,u z ) = 0 , (2.14)2 u ( F − F ) + u z ( F ,u F ,S z − F ,u F ,S z ) − F ,u = 0 , (2.15) F F ,S z + F ,u z = 0 , (2.16)2 u ( F − F ) + u z ( F ,u F ,S z − F ,u F ,S z ) + F ,u = 0 , (2.17)( u + 1)( F − F ) + u z ( F ,S z F ,u z − F ,S z F ,u z ) = 0 , (2.18)( u + 1) ( F − F ) + u z ( F ,S z F ,u z − F ,S z F ,u z ) = 0 , (2.19)2 u ( u + 1)( F − F ) + u z ( F ,S z F ,u − F ,S z F ,u ) = 0 . (2.20)Now we will work with Eqs. (2.10)–(2.12) to clarify functions F , F . Let us express F ,u z , F ,u z from (2.10) and substitute them into (2.12). This leads to the condition F = F or to theequation (cid:0) − u z F ,S z F ,S z (cid:1) = 0 . (2.21)Let us consider case (2.21). We look for F , F in the following form: F ( u, u z , S z ) = A ( u, S z ) u z , F ( u, u z , S z ) = B ( u, S z ) u z . (2.22)Then A , B have to satisfy the system obtained using (2.21), (2.10), and (2.11),1 − A S z B S z = 0 , − A + BA S z = 0 , − B + AB S z = 0 , (2.23) B u − A u + B S z A u − A S z B u = 0 . (2.24)This system has the solution: A ( u, S z ) = e a ( u ) S z + a ( u ) a ( u ) − a ( u ) . Here a , a are arbitrary functions. Similarly, we find that B ( u, S z ) = e a ( u ) S z + a ( u ) a ( u ) − a ( u )with arbitrary functions a , a . Under obtained A and B the first equation (2.23) becomes1 − e ( a ( u )+ a ( u )) S z + a ( u ) a ( u )+ a ( u ) a ( u ) = 0 . Thus one can derive that a = − a , a = a . Finally, equation (2.24) takes the form (cid:0) − a ( u ) a ′ ( u ) S z − a ( u ) a ′ ( u ) − a ( u ) a ( u ) a ′ ( u ) + 2 a ′ ( u ) (cid:1) e a ( u )( S z + a ( u )) ++ (cid:0) a ( u ) a ′ ( u ) + a ( u ) a ( u ) a ′ ( u ) + 2 a ′ ( u ) + a ( u ) a ′ ( u ) S z (cid:1) e − a ( u )( S z + a ( u )) − a ′ ( u ) = 0 . (2.25)We assume essential dependence on S z for functions F , F and, therefore, for A , B , so thefunctions e a ( u ) S z , e − a ( u ) S z , e a ( u ) S z S z , e − a ( u ) S z S z are independent. Hence we have a ( u ) = c , a ( u ) = c , where c , c are arbitrary constants. M.N. KuznetsovaThus, we have clarified the right hand sides of Lax pair (2.2), (2.3) S x = F ( u, u x , u y , u z , S z ) = (cid:0) e c ( S z + c ) − (cid:1) u x c u z + F ( u, u z , S z ) ,S y = G ( u, u x , u y , u z , S z ) = − (cid:0) e − c ( S z + c ) − (cid:1) u y c u z + F ( u, u z , S z ) . By the shift transformation S → S − c z and by the scaling z → c z these equations can bereduced to S x = F ( u, u x , u y , u z , S z ) = (cid:0) e S z − (cid:1) u x u z + F ( u, u z , S z ) ,S y = G ( u, u x , u y , u z , S z ) = − (cid:0) e − S z − (cid:1) u y u z + F ( u, u z , S z ) . To clarify F , we substitute the above functions into (2.13), (2.14), and (2.15)( e − S z − u z F ,u − u ( e S z + 2) = 0 ,u z ( e − S z − − u z e − S z ) F ,S z − ( u + 1)( e S z + e − S z −
2) = 0 , − ( e − S z − F ,S z + u z F ,u z = 0 . This system has the solution: F ( u, u z , S z ) = − ( e Sz − u +1) u z . Now we rewrite Eqs. (2.16)–(2.20)and we obtain the system on the unknown function F :( e S z − F ,S z + u z F ,u z = 0 ,u z ( e S z − F ,S z + u z e S z F ,u z − ( u + 1)( e S z + e − S z −
2) = 0 , − u z ( − e − S z + 3 e − S z − e S z ) F ,S z − u z ( e S z + e − S z − F ,u z ++( u + 1)( − e S z − e − S z + 6 e − S z + e − S z ) = 0 ,u z (1 − e S z ) F ,u − u ( e S z + e − S z −
2) = 0 , uu z ( e − S z − e − S z + 3 − e S z ) F ,S z + ( u + 1) u z ( e S z + e − S z − F ,u ++2 u ( u + 1)( e − S z + 6 e − S z − e − S z + e S z −
4) = 0 . This system possesses the solution F ( u, u z , S z ) = − (1 − e − Sz )( u +1) u z .Thus we have found the Lax pair S x = u x − u − u z ( e S z −
1) + 1 u z , (2.26) S y = u y − u − u z (1 − e − S z ) − u z . (2.27)for equation (2.1).Now we reconstruct the dispersive Lax pair by an appropriate quantization the dispersionlessLax pair (2.26), (2.27). First, we “quantise” [8] the first term in every equation (2.26), (2.27): u z is replaced by △ z u ; e S z − △ z ψ due to the formal representation e ∂∂z ≈ ∂∂z + · · · , and,similarly 1 − e − S z by △ ¯ z ψ . Note that if we replace u z by △ z u in the second term in the r.h.s ofequations (2.26), (2.27), then obtained system will not be the Lax pair for (1.8). That is whywe fit this term by the following way: ψ x = u x − u − △ z u △ z ψ + P ( u ) ψ,ψ y = u y − u − △ ¯ z u △ ¯ z ψ + Q ( u ) ψ. ax pair for one novel two-dimensional lattice 7The compatibility condition ψ xy = ψ yx is straightforward to solve. Thus we find that equation(1.8) possesses the Lax pair ψ x = u x − u − △ z u △ z ψ + uψ,ψ y = u y − u − △ ¯ z u △ ¯ z ψ + uψ. It finally proved Theorem 2.1. (cid:4)
Let us impose the periodic closure conditions u n +2 = u n to infinite lattice (1.4). Then we obtainthe following finite system: u ,xy = 2 u − u u ,x u ,y , u ,xy = 2 u − u u ,x u ,y . (3.1)System (3.1) has the x -integral and the y -integral w = u ,y u ,y ( u − u ) , W = u ,x u ,x ( u − u ) . (3.2)Lax pair for (3.1) has the form:Ψ x = ( Aλ + B )Ψ , Ψ y = ( ˜ Aλ − + ˜ B )Ψ , (3.3)where Ψ = ( ψ , ψ ) T and A = (cid:18) u ,x u − u (cid:19) , B = (cid:18) − u ,x u − u u ,x u − u − u ,x u − u (cid:19) , (3.4)˜ A = (cid:18) − u ,y u − u (cid:19) , ˜ B = (cid:18) − u ,y u − u − u ,y u − u − u ,y u − u (cid:19) , (3.5) λ is a spectral parameter.The classical symmetry can be found directly from the consistency condition( u i,xy ) t = ( u i,t ) xy , u ,t = ( u − u ) u ,x F ( W ) + c u − u ) u ,x + c u + c u ,u ,t = ( u − u ) u ,x F ( W ) + c u − u ) u ,x + c u + c u , where F - is an arbitrary function depending on the y -integral W defined by the first of formulas(3.2); c i are arbitrary constants. The classical symmetry in the another direction is simply foundbecause the system is symmetric under the change of variables x ↔ y : u ,τ = ( u − u ) u ,y G ( w ) + ˜ c u − u ) u ,y + ˜ c u + ˜ c u ,u ,τ = ( u − u ) u ,y G ( w ) + ˜ c u − u ) u ,y + ˜ c u + ˜ c u . Higher symmetry of the second order is seeked in the following form: u i,t = a i ( u , u , u ,x , u ,x ) u ,xx + b i ( u , u , u ,x , u ,x ) u ,xx + h i ( u , u , u ,x , u ,x ) , i = 1 , , (3.6) M.N. Kuznetsovawhere a i , b i , h i are functions to be found. To find the higher symmetry we use Lax pair (3.3).Let us consider the linear problemΨ t = ( αλ + βλ + γ )Ψ , (3.7)where α = ( α i,j ), β = ( β i,j ), γ = ( γ i,j ), i, j = 1 , u , u , u ,x , u ,x , u ,xx , u ,xx . The compatibilitycondition (Ψ x ) t = (Ψ t ) x for the systemsΨ x = ( Aλ + B )Ψ , Ψ t = ( αλ + βλ + γ )Ψ , results in the system of relations: Aα = αA, Aβ + Bα = α x + αB + βA,A t + Aγ + Bβ = β x + βB + γA, B t + Bγ = γ x + γB. A complete study of these equations leads to the following formulas: u ,t = H ( W ) u ,xx + u ,x ( u − u ) Φ( W ) u ,xx + ( u − u ) g ( u , u , u ,x , u ,x ) + (3.8)+( u − u )( c − c u − c ) − ( c u + c u + c ) , (3.9) u ,t = u ,x u ,x H ( W ) u ,xx + W Φ( W ) u ,xx + ( u − u ) u ,x u ,x g ( u , u , u ,x , u ,x ) + (3.10)+ ( u − u ) u ,x u ,x ( c + c u + c ) − ( c u + c u + c ) , (3.11)where H, Φ , g are arbitrary functions; c i are arbitrary constants. To define precisely obtainedformulas we substitute them into the compatibility condition ( u i,xy ) t = ( u i,t ) xy . Thus, we finallyfound the higher symmetry: u ,t = W F ( W ) u ,xx + u ,x ( u − u ) F ( W ) u ,xx + − u ,x u ,x ( u ,x − u ,x )( u − u ) F ( W ) − u ,x G ( W ) − ξ u − ξ u − ξ (3.12) u ,t = u ,x ( u − u ) u ,xx F ( W ) + W F ( W ) u ,xx + − u ,x u ,x ( u ,x − u ,x )( u − u ) F ( W ) − u ,x G ( W ) − ξ u − ξ u − ξ , (3.13)where F , G are arbitrary functions; W is the y -integral defined by (3.2); ξ i are arbitrary con-stants. Note that if we set F ( W ) = 0 then we obtain the classical symmetry. Also we finallyfound matrices α , β , γ involved in (3.7): α = (cid:18) α α (cid:19) , β = (cid:18) β β ( u, u x , u xx ) β (cid:19) ,γ = (cid:18) γ ( u, u x , u xx ) γ ( u, u x , u xx )0 γ ( u, u x , u xx ) (cid:19) , where β ( u, u x , u xx ) = u ,x ( u − u ) F ( W ) u ,xx + u ,x u ,x ( u − u ) F ( W ) u ,xx −− u ,x u ,x ( u ,x − u ,x )( u − u ) F ( W ) − u ,x u − u G ( W ) , ax pair for one novel two-dimensional lattice 9 γ ( u, u x , u xx ) = u ,x u ,x ( u − u ) F ( W ) u ,xx + u ,x ( u − u ) F ( W ) u ,xx −− u ,x u ,x ( u ,x − u ,x )( u − u ) F ( W ) − u ,x u − u G ( W ) − ξ u − ξ ,γ = − u ,x u ,x ( u − u ) F ( W ) u ,xx − u ,x ( u − u ) F ( W ) u ,xx ++ u ,x u ,x ( u ,x − u ,x )( u − u ) F ( W ) + u ,x u − u G ( W ) ,α , β are arbitrary constants. Thus it is seen that definitive answer is given by formulas(3.12), (3.13) andΨ t = ( βλ + γ )Ψ , β = (cid:18) β (cid:19) , γ = (cid:18) γ γ γ (cid:19) , where β , γ ij have been described just above.Let us consider chain (1.6). We impose the periodic closure conditions u n +2 = u n to infinitechain (1.6) and obtain the following finite system: u ,xy = u − u ( u ,x − u − u ,y − u −
1) + 2 u ( u ,x + u ,y − u − ,u ,xy = u − u ( u ,x − u − u ,y − u −
1) + 2 u ( u ,x + u ,y − u − . (3.14)This system possesses the y -integral and x -integral W = ( u ,x − u − u ,x − u − u − u ) , w = ( u ,y − u − u ,y − u − u − u ) . (3.15)System (3.14) is the compatibility condition for the Lax pairΨ x = ( Aλ + B )Ψ , Ψ y = ( ˜ Aλ − + ˜ B ) , (3.16)where Ψ = ( ψ , ψ ) T , A = u ,x − u − u − u ! , B = − u ,x − u − u − u + u u ,x − u − u − u − u ,x − u − u − u + u ! , ˜ A = − u ,y − u − u − u ! , ˜ B = u ,y − u − u − u + u − u ,y − u − u − u u ,y − u − u − u + u ! . To find the higher symmetry it is sufficient (as we have just seen) to consider the systemΨ t = ( βλ + γ )Ψ , (3.17)compatible with the first equation of (3.16). In this way we obtained the higher symmetry ofsystem (3.14): u ,t = F ( W − u ,xx + − u ,x + u + 1( u − u ) W F ( W − u ,xx − ( u ,x − u − G ( W ) −− φ ( u , u , u ,x , u ,x )( − u ,x + u + 1)( u − u ) F ( W − − c ( u + 1) , u ,t = ( u − u ) ( − u ,x + u + 1) W F ( W − u ,xx + F ( W − u ,xx + ( u − u ) W ( − u ,x + u + 1) G ( W − −− φ ( u , u , u ,x , u ,x ) F ( W − − u ,x + u + 1) − c ( u + 1) , where W is the y -integral given by (3.15), ϕ ( u , u , u ,x , u ,x ) = u ,x u ,x ( u ,x − u ,x ) + u ,x (1 + u ) − u ,x (1 + u )+ − u ,x (1 + u + u u + u u ) + u ,x (1 + u + u u + u u ) . (3.18)Matrices β , γ (see (3.7)) are defined by the following formulas: β = (cid:18) β (¯ u, ¯ u x , ¯ u xx ) 0 (cid:19) , γ = (cid:18) γ (¯ u, ¯ u x , ¯ u xx ) γ (¯ u, ¯ u x , ¯ u xx )0 γ (¯ u, ¯ u x , ¯ u xx ) (cid:19) , where β (¯ u, ¯ u x , ¯ u xx ) = − u ,x + u + 1( u − u )( − u ,x + u + 1) F ( W − u ,xx + F ( W − u − u u ,xx −− ϕ ( u , u , u ,x , u ,x ) F ( W − − u ,x + u + 1)( u − u ) + − u ,x + u + 1 u − u G ( W − ,γ (¯ u, ¯ u x , ¯ u xx ) = F ( W − u ,xx u − u + ( − u ,x + u + 1) F ( W − u ,xx ( − u ,x + u + 1)( u − u ) ++ ( − u ,x + u + 1) G ( W − u − u ) − ϕ ( u , u , u ,x , u ,x )( u − u ) ( − u ,x + u + 1) F ( W − − c − c u ,γ (¯ u, ¯ u x , ¯ u xx ) = − F ( W − u ,xx u − u − ( − u ,x + u + 1) F ( W − − u ,x + u + 1)( u − u ) u ,xx −− ( − u ,x + u + 1) G ( W − u − u + 2 ϕ ( u , u , u ,x , u ,x ) F ( W − u − u ) ( − u ,x + u + 1) ,γ (¯ u, ¯ u x , ¯ u xx ) = − ( − u ,x + u + 1) F ( W − u − u )( − u ,x + u + 1) u ,xx − F ( W − u ,xx u − u −− ( − u ,x + u + 1) u − u G ( W −
1) + 2 ϕ ( u , u , u ,x , u ,x )( − u ,x + u + 1)( u − u ) F ( W − − c − c u ,ϕ ( u , u , u ,x , u ,x ) is defined by (3.18), c , c are arbitrary constants.Note, that periodic closing obtained by the conditions u n +3 = u n imposing on infinite chain(1.6) leads to the system u ,xy = (cid:18) u − u − u − u (cid:19) ( u ,x − u − u ,y − u −
1) + 2 u ( u ,x + u ,y − u − ,u ,xy = (cid:18) u − u − u − u (cid:19) ( u ,x − u − u ,y − u −
1) + 2 u ( u ,x + u ,y − u − ,u ,xy = (cid:18) u − u − u − u (cid:19) ( u ,x − u − u ,y − u −
1) + 2 u ( u ,x + u ,y − u − . ax pair for one novel two-dimensional lattice 11This system has y -integral and x -integral W = ( u ,x − u − u ,x − u − u ,x − u − u − u )( u − u )( u − u ) ,w = ( u ,y − u − u ,y − u − u ,y − u − u − u )( u − u )( u − u ) . Lax pair has the following form:Ψ x = ( Aλ + B )Ψ , Ψ y = ( ˜ Aλ − + ˜ B ) , where Ψ = ( ψ , ψ , ψ ) T , A = u ,x − u − u − u , B = − u ,x − u − u − u + u u ,x − u − u − u − u ,x − u − u − u + u u ,x − u − u − u − u ,x − u − u − u + u , ˜ A = − u ,y − u − u − u , ˜ B = u ,y − u − u − u + u − u ,y − u − u − u u ,y − u − u − u + u − u ,y − u − u − u u ,y − u − u − u + u . The problem of classification multidimensional equations is actively studied by many authors,using different algebraic and geometry approaches [15, 17, 18, 19, 20, 21, 22, 23, 24]. We notethat the classification algorithm for integrable two-dimensional lattices proposed in our previouspapers does not provide any algorithm for constructing the Lax pair.It is known that finite systems obtained from infinite integrable chains by degenerate bound-ary conditions imposing at the two points of the form u n + k = c , u n + s = c (where c , c areconstants) are integrable in the sense of Darboux (they have complete set of integrals in bothcharacteristic directions, i.e. the number of independent integrals is equal to the order of thesystem). We proved that finite systems obtained from infinite chains (1.4), (1.6) by periodicclosure conditions possess higher symmetries. Actually they are integrable. It is interestingfact that each of these systems also has one x -integral and one y -integral. It is not sufficientfor Darboux integrability but it is the unusual feature of the system. So a new problem hasemerged to study characteristic algebras of the systems with an incomplete set of integrals. Ina discrete version, this problem is discussed in paper [25]. Acknowledgements
The author thanks I. T. Habibulin for assignment the problem and useful discussions andE. V. Ferapontov for explaning the method of the construction of Lax pairs.
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