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Continuous symmetries of difference equations. (English) Zbl 1112.37053
Summary: Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict them to point symmetries, but to allow them to also transform the lattice.
MSC:
37K05Hamiltonian structures, symmetries, variational principles, conservation laws
35A30Geometric theory for PDE, characteristics, transformations
39A11Stability of difference equations (MSC2000)
39A12Discrete version of topics in analysis
65Q05Numerical methods for functional equations (MSC2000)
81R12Relations of groups and algebras in quantum theory with integrable systems