On the classification of Darboux integrable chains
Date
2008Source Title
Journal of Mathematical Physics
Print ISSN
00222488
Electronic ISSN
10897658
Publisher
American Institute of Physics
Volume
49
Issue
10
Pages
1027021  10270239
Language
English
Type
ArticleItem Usage Stats
167
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174
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Abstract
We study differentialdifference 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 EquationsDifferentiation
Integral equations
Lie algebras
Mathematical operators
Difference equations
Differentiation
Integral equations
Lie Algebras
Mathematical operators
Equations
Permalink
http://hdl.handle.net/11693/11637Published Version (Please cite this version)
http://dx.doi.org/10.1063/1.2992950Collections
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