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

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      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
      Article
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      115
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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.
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      http://hdl.handle.net/11693/22571
      Published Version (Please cite this version)
      http://dx.doi.org/10.1063/1.3251334
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      • Department of Mathematics 635
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