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

HABIBULLIN I. , Zheltukhina N., Pekcan A.

JOURNAL OF MATHEMATICAL PHYSICS, vol.50, no.10, 2009 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 50 Issue: 10
  • Publication Date: 2009
  • Doi Number: 10.1063/1.3251334
  • Keywords: difference equations, differentiation, integral equations, Lie algebras, mathematical operators, EQUATIONS


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=-infinity)(infinity), {(d(k)/dx(k))t(n,x)}(k=1)(infinity), such that DxF=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).