Complete list of Darboux integrable chains of the form t1x=tx+d(t,t1)

Abstract

We study differential-difference equation (d/dx)t(n+1,x)=f(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. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, {t(n+k,x)}k=−∞∞, {(dk/dxk)t(n,x)}k=1∞, such that DxF=0 and DI=I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp(n)=p(n+1). Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function f is of the special form f(u,v,w)=w+g(u,v).