Variational formulation of approximate symmetries and conservation laws. (English) Zbl 1006.81035

Johnpillai, A. G.; Kara, A. H.

Int. J. Theor. Phys. 40, No. 8, 1501-1509 (2001).

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.

Reviewer: Zhenghan Wang (Bloomington)

MSC:

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

Keywords:

approximate Lie symmetry generators of a partial differential equation; approximate Noether symmetries