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Applications of Lie Groups to Difference Equations
Preface Introduction Brief introduction to Lie group analysis of differential equations Preliminaries: Heuristic approach in examples Finite Differences and Transformation Groups in Space of DiscreteExpand
Invariance and first integrals of continuous and discrete Hamiltonian equations
The relation between symmetries and first integrals for both continuous canonical Hamiltonian equations and discrete Hamiltonian equations is considered. The observation that canonical HamiltonianExpand
Noether-type theorems for difference equations
Abstract The Noether's type constructions for difference functionals, difference equations and meshes (lattices) are reviewed. It is shown in Dorodnitsyn [J. Soviet Math. 55 (1999) 1490]; [Dokl.Expand
Continuous symmetries of Lagrangians and exact solutions of discrete equations
One of the difficulties encountered when studying physical theories in discrete space–time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of theExpand
Symmetry-adapted moving mesh schemes for the nonlinear Schrodinger equation
In this paper we consider symmetry-preserving difference schemes for the nonlinear Schrodinger equation where n is the number of space dimensions. This equation describes one-dimensional waves in nExpand
Finite Difference Models Entirely Inheriting Continuous Symmetry Of Original Differential Equations
The present paper is concerned with continuous groups of transformations in a space of discrete variables. The criterion of invariance of difference equations together with difference grid isExpand
A quasilinear heat equation with a source: Peaking, localization, symmetry exact solutions, asymptotics, structures
A survey is given of results of investigating unbounded solutions (regimes with peaking) of quasilinear parabolic equations of nonlinear heat conduction with a source. Principal attention is devotedExpand
Transformation groups in net spaces
We consider formal groups of transformations on the space of differential and net (finite-difference) variables. We show that preservation of meaning of difference derivatives under transformationsExpand
Symmetry-preserving difference schemes for some heat transfer equations
Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristicExpand