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Variational formulation of approximate symmetries and conservation laws. (English) Zbl 1006.81035
T. Feroze and A. H. Kara [Int. J. Non-Linear Mech. 37, No. 2, 275–280 (2002)] have shown that the procedure to construct Lagrangians for differential equations, using approximate symmetries and associated conservation laws along with Noether’s theorem, can be extended to ordinary differential equations with a small parameter. In the paper under review, the authors extend the procedure to perturbed partial differential equations. They show how the conserved vectors and associated approximate Lie symmetry generators of a partial differential equation with a small parameter can be utilized to construct approximate Noether symmetries and hence, new associated conservation laws. The theory is applied to a number of perturbations of the wave equation.

MSC:
81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
22E70 Applications of Lie groups to the sciences; explicit representations
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