Infinite Conservation Laws for Infinite Symmetries

Abstract:
We will consider differential equations possessing infinite symmetry algebras and infinite sets of local conservation laws.

Infinite symmetry algebras with arbitrary functions of independent variables and their corresponding conservation laws were studied in V.Rosenhaus, "Infinite Symmetries and Conservation Laws", J. Math. Phys. 43, 6129-6150 (2002)]. It was shown that these symmetries may lead to a finite number of local conservation laws that are determined by a specific form of boundary conditions. Conservation laws corresponding to infinite symmetries were calculated for a number of interesting equations, e.g.: V.Rosenhaus, "On conservation laws for the equation of non-stationary transonic gas flows", J. Dyn. Sys. Geom. Theor. 1, 95-107 (2002), "On conservation laws and boundary conditions for short waves equation", Rep. Math. Phys. 51, 71-86 (2003), "On infinite symmetries and essential conservation laws for Navier-Stokes equations", Proc. XXIV Intern. Conf. Group Theor. Methods in Physics (Paris, July, 2002), pp.737-740.

The present work is an extension of this approach for Lagrangian differential equations whose symmetry algebras contain arbitrary functions of dependent variables and their derivatives. We will show that these symmetries lead to an infinite number of local conservation laws. We will discuss classification, boundary conditions and give examples.