A method of integrable discretization of the Liouville type nonlinear partial differential equations is suggested based on integrals. New examples of discrete Liouville type models are presented.

Differential-difference equation d dx t(n + 1, x) = f (x, t(n, x), t(n + 1, x), d dx t(n, x)) with unknown t(n, x) depending on continuous and discrete variables x and n is studied. We call an equation of such kind Darboux integrable, if there exist two functions (called integrals) F and I of a finite number of dynamical variables such that D x F = 0 and DI = I, where D x is the operator of total differentiation with respect to x, and D is the shift operator: Dp(n) = p(n+1).It is proved that the integrals can be brought to some canonical form. A method of construction of an explicit formula for general solution to Darboux integrable chains is discussed and for a class of chains such solutions are found.1 two decades the discrete phenomena have become very popular due to various important applications (for more details see [1]-[3] and references therein).Below we use a subindex to indicate the shift of the discrete argument: t k = t(n+k, x), k ∈ Z, and derivatives with respect to x: t [1] We denote through D and D x the shift operator and the operator of the total derivative with respect to x correspondingly. For instance, Dh(n, x) = h(n + 1, x) and D x h(n, x) = d dx h(n, x). Functions I and F , both depending on x, n, and a finite number of dynamical variables, are called respectively n-and x-integrals of (1), if DI = I and D x F = 0 (see also [4]). Clearly, any function depending on n only, is an x-integral, and any function, depending on x only, is an n-integral. Such integrals are called trivial integrals. One can see that any n-integral I does not depend on variables t m , m ∈ Z\{0}, and any x-integral F does not depend on variables t [m] , m ∈ N.Chain (1) is called Darboux integrable if it admits a nontrivial n-integral and a nontrivial xintegral.The basic ideas on integration of partial differential equations of the hyperbolic type go back to classical works by Laplace, Darboux, Goursat, Vessiot, Monge, Ampere, Legendre, Egorov, etc.Notice that understanding of integration as finding an explicit formula for a general solution was later replaced by other, in a sense less obligatory, definitions. For instance, the Darboux method for integration of hyperbolic type equations consists of searching for integrals in both directions followed by the reduction of the equation to two ordinary differential equations. In order to find integrals, provided that they exist, Darboux used the Laplace cascade method. An alternative, more algebraic approach based on the characteristic vector fields was used by Goursat and Vessiot. Namely this method allowed Goursat to get a list of integrable equations [5]. An important contribution to the development of the algebraic method investigating Darboux integrable equations was made by A.B. Shabat who introduced the notion of the characteristic algebra of the hyperbolic equation u x,y = f (x, y, u, u x , u y ) .(2)It turned out that the operator D y of total differentiation, with respect to the variable y, defines a derivative in the characteristic algebra in the direction of x. Mo...

In the article some algebraic properties of nonlinear two-dimensional lattices of the form u n,xy = f (u n+1 , u n , u n−1 ) are studied. The problem of exhaustive description of the integrable cases of this kind lattices remains open. By using the approach, developed and tested in our previous works we adopted the method of characteristic Lie-Rinehart algebras to this case. In the article we derived an effective integrability conditions for the lattice and proved that in the integrable case the function f (u n+1 , u n , u n−1 ) is a quasi-polynomial satisfying the following equationwhere C and α are constant parameters and k, m are nonnegative integers.

In the article some algebraic properties of nonlinear two-dimensional lattices of the form u n,xy = f (u n+1 , u n , u n−1 ) are studied. The problem of exhaustive description of the integrable cases of this kind lattices remains open. By using the approach, developed and tested in our previous works we adopted the method of characteristic Lie-Rinehart algebras to this case. In the article we derived an effective integrability conditions for the lattice and proved that in the integrable case the function f (u n+1 , u n , u n−1 ) is a quasi-polynomial satisfying the following equation ∂ 2 ∂un+1∂un−1 f (u n+1 , u n , u n−1 ) = Ce αun− αm 2 un+1− αk 2 un−1 , where C and α are constant parameters and k, m are nonnegative integers.