## Complete list of Darboux integrable chains of the form t 1 x = t x + d ( t, t 1 )

##### Author

Habibullin, I.

Zheltukhina, N.

Pekcan, A.

##### Date

2009##### Source Title

Journal of Mathematical Physics

##### Print ISSN

0022-2488

##### Volume

50

##### Issue

10

##### Pages

102710-1 - 102710-23

##### Language

English

##### Type

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##### 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 /d xk) t (n,x) } k=1 ∞, such that Dx 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). 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). © 2009 American Institute of Physics.