Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras

Abstract

We consider a classification problem of integrable cases of the Toda type twodimensional lattices , = ( +1 , , −1 , , , , ). The function = ( 1 , 2 , · · · 5 ) is assumed to be analytic in a domain ⊂ C 5 . The sought function = ( , ) depends on real , and integer . Equations with three independent variables are complicated objects for study and especially for classification. It is commonly accepted that for a given equation, the existence of a large class of integrable reductions indicates integrability. Our classification algorithm is based on this observation. We say that a constraint 0 = ( , ) defines a degenerate cutting off condition for the lattice if it divides this lattice into two independent semi-infinite lattices defined on the intervals −∞ < < 0 and 0 < < +∞, respectively. We call a lattice integrable if there exist cutting off boundary conditions allowing us to reduce the lattice to an infinite number of hyperbolic type systems integrable in the sense of Darboux. Namely, we require that lattice is reduced to a finite system of such kind by imposing degenerate cutting off conditions at two different points = 1 , = 2 for arbitrary pair of integers 1 , 2 . Recall that a system of hyperbolic equations is called Darboux integrable if it admits a complete set of integrals in both characteristic directions. An effective criterion of the Darboux integrability of the system is connected with properties of an associated algebraic structures. More precisely, the characteristic Lie-Rinehart algebras assigned to both characteristic directions have to be of a finite dimension. Since the obtained hyperbolic system is of a very specific form, the characteristic algebras are effectively studied. Here we focus on a subclass of quasilinear lattices of the form , = ( −1 , , +1 ) , + ( −1 , , +1 ) , + ( −1 , , +1 ).

Lax Pair for a Novel Two-Dimensional Lattice

“…In a number of recent publications [8,13,14,15,16,18] the problem of integrable classification of two-dimensional lattices u n,xy = f (u n+1 , u n , u n−1 , u n,x , u n,y ), −∞ < n < ∞, (1.1) was studied. Here the sought function u n = u n (x, y) depends on the real variables x, y and the integer variable n. In these papers we proposed the method for seeking and classifying integrable equations with three independent variables based on the requirement of the existence of a set of Darboux integrable reductions and on the notion of the characteristic Lie-Rinehart algebras.…”

Lax Pair for a Novel Two-Dimensional Lattice

“…A number of our recent publications [1,2,3,4,5,6] are addressed the problem of integrable classification of two-dimensional lattices…”

“…This method is based on the requirement of the existence of an infinite set of Darboux integrable reductions and on the notion of the characteristic Lie-Rinehart algebras. The method was applied for the classification of integrable cases of different subclasses of equations [3,4,5,6]. Under this approach the novel integrable chainwas obtained.…”

On a classification algorithm of the integrable two-dimensional lattices via Lie-Rinehart algebras

“…When such kind boundary conditions are imposed at two different points n = N 1 and n = N 2 then the lattice reduces to a Darboux integrable system of the hyperbolic type equations. We suggested and developed in our works [22]- [26] a classification algorithm based on this observation. Let's briefly discuss the essence of the method.…”

“…In the case when both of these requirements are violated the lattice can be reduced to the form un,xy = βun,x + γun,y + δ.On the classification of this kind lattices see[26].…”