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      On the classification of Darboux integrable chains

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      Author
      Habibullin, I.
      Zheltukhina, N.
      Pekcan, A.
      Date
      2008
      Source Title
      Journal of Mathematical Physics
      Print ISSN
      0022-2488
      Electronic ISSN
      1089-7658
      Publisher
      American Institute of Physics
      Volume
      49
      Issue
      10
      Pages
      102702-1 - 102702-39
      Language
      English
      Type
      Article
      Item Usage Stats
      101
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      82
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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.
      Keywords
      Difference Equations
      Differentiation
      Integral equations
      Lie algebras
      Mathematical operators
      Difference equations
      Differentiation
      Integral equations
      Lie Algebras
      Mathematical operators
      Equations
      Permalink
      http://hdl.handle.net/11693/11637
      Published Version (Please cite this version)
      http://dx.doi.org/10.1063/1.2992950
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