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

J. Math. Phys. 50, 102710 (2009); doi:10.1063/1.3251334

Published 30 October 2009

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Ismagil Habibullin,1 Natalya Zheltukhina,2 and Aslı Pekcan2
1Ufa Institute of Mathematics, Russian Academy of Science, Chernyshevskii Str., 112, Ufa 450077, Russia
2Department of Mathematics, Faculty of Science, Bilkent University, 06800 Ankara, Turkey
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)}<sub>k = -[infinity]</sub><sup>[infinity]</sup>, {(dk/dxk)t(n,x)}<sub>k = 1</sub><sup>[infinity]</sup>, 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). ©2009 American Institute of Physics
History: Received 28 July 2009; accepted 23 September 2009; published 30 October 2009
Permalink: http://link.aip.org/link/?JMAPAQ/50/102710/1
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0022-2488 (print)   1089-7658 (online)
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