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Symmetry groups of differential–difference equations and their compatibility

Authors
Journal
Journal of Mathematical Analysis and Applications
0022-247X
Publisher
Elsevier
Publication Date
Volume
371
Issue
1
Identifiers
DOI: 10.1016/j.jmaa.2010.05.025
Keywords
  • Symmetry
  • Differential–Difference Equation
  • Compatibility
  • Bäcklund Transformation

Abstract

Abstract It is shown that the intrinsic determining equations of a given differential–difference equation (DDE) can be derived by the compatibility between the original equation and the intrinsic invariant surface condition. The ( 2 + 1 ) -dimensional Toda lattice, the special Toda lattice and the DD-KP equation serving as examples are used to illustrate this approach. Then, Bäcklund transformations of the ( 2 + 1 ) -dimensional DDEs including the special Toda lattice, the modified Toda lattice and the DD-KZ equation are presented by using the non-intrinsic direct method. In addition, the Clarkson–Kruskal direct method is developed to find similarity reductions of the DDEs.

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