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      On Darboux-integrable semi-discrete chains

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      Author(s)
      Habibullin, I.
      Zheltukhina, N.
      Sakieva, A.
      Date
      2010
      Source Title
      Journal of Physics A: Mathematical and Theoretical
      Print ISSN
      17518113
      Volume
      43
      Issue
      43
      Pages
      434017
      Language
      English
      Type
      Article
      Item Usage Stats
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      Abstract
      A differential-difference equation d/dx t(n + 1, x) = f (x, t(n, x), t(n + 1, x), d/dx t(n, x)) with unknown t(n, x) depending on the continuous and discrete variables x and n is studied. We call an equation of such kind Darboux integrable if there exist two functions (called integrals) F and I of a finite number of dynamical variables 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). It is proved that the integrals can be brought to some canonical form. A method of construction of an explicit formula for a general solution to Darboux-integrable chains is discussed and such solutions are found for a class of chains. © 2010 IOP Publishing Ltd.
      Permalink
      http://hdl.handle.net/11693/22164
      Published Version (Please cite this version)
      http://dx.doi.org/10.1088/1751-8113/43/43/434017
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