On conservation laws for infinite symmetries

Abstract:
We consider partial differential equations possessing infinite symmetry algebras parametrized by arbitrary functions of independent variables of the problem as well as the case of dependent variables and their derivatives. We discuss associated local conservation laws with non-vanishing conserved densities and the role of corresponding boundary conditions.

It was shown that infinite symmetry algebras with arbitrary functions of (not all) independent variables may lead to a finite number of conserved densities that are determined by a specific form of boundary conditions (V.Rosenhaus, J. Math. Phys. 43, 6129 (2002)). Using this approach conservation laws corresponding to infinite symmetries were calculated for various systems, incl. the equation of transonic gas flows, short waves equation, Zabolotskaya-Khokhlov equation, Kadomtsev-Petviashvili equation. We discuss how to generate conserved densities and boundary conditions corresponding to infinite symmetry algebra of the Davey-Stewartson equations (joint work with M. Gandarias).

Unlike infinite symmetries with arbitrary functions of independent variables arbitrary functions of dependent variables and their derivatives lead to infinite number of conservation laws (V.Rosenhaus, Theor. Math. Phys. 144, 1046 (2005)). We discuss differential equations of the second order admitting infinite variational symmetries with arbitrary functions of a dependent variable as well as its first derivatives. We give examples of classes of such systems and corresponding infinite conserved quantities (V.Rosenhaus, Theor. Math. Phys. 2007).