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Lagrangian approach to evolution equations: symmetries and conservation laws. (English) Zbl 1106.70012
This paper describes a method to derive conservation laws for some evolution equations. Given an evolution equation non admitting a Lagrangian description, then, if there exists a new evolution equation in a new dependent variable or field in such a way that one can find a Lagrangian formulation for the coupled system of evolution equations, the study of Lagrangian symmetries for the new system can be used in order to obtain conservation laws, via Noether’s theorem, for the original problem. By using this method, the authors give Lagrangian descriptions of heat equation, Burgers equation, nonlinear heat equation, and nonlinear Schrödinger and Korteweg-de Vrieg type systems. As a first example, the infinite set of known conservation laws of heat equation is described by applying Noether’s theorem. Next, the method is applied to the nonlinear heat equation, as well as to Burgers equation, and new non-local conservation laws are obtained.
MSC:
70H33Symmetries and conservation laws, reverse symmetries, invariant manifolds, etc.
70G65Symmetries, Lie-group and Lie-algebra methods for dynamical systems
35Q53KdV-like (Korteweg-de Vries) equations
35Q55NLS-like (nonlinear Schrödinger) equations