Characteristic lie algebra and classification of semi-discrete models

Abstract

In this thesis, we studied a differential-difference equation of the following form tx(n + 1, x) = f(t(n, x), t(n + 1, x), tx(n, x)), (1) where the unknown t = t(n, x) is a function of two independent variables: discrete n and continuous x. The equation (1) is called a Darboux integrable equation if it admits nontrivial x- and n-integrals. A function F(x, t, t±1, t±2, ...) is called an x-integral if DxF = 0, where Dx is the operator of total differentiation with respect to x. A function I(x, t, tx, txx, ...) is called an n-integral if DI = I, where D is the shift operator: Dh(n) = h(n + 1). In this work, we introduced the notion of characteristic Lie algebra for semidiscrete hyperbolic type equations. We used characteristic Lie algebra as a tool to classify Darboux integrability chains and finally gave the complete list of Darboux integrable equations in the case when the function f in the equation (1) is of the special form f = tx(n, x) + d(t(n, x), t(n + 1, x)).