dr Roman Cherniha (Kijów)

The Galilean relativistic principle and nonlinear partial differential equations

The Galilean invariance of nonlinear partial differential equations (PDEs) is studied. We start from the standard representation of Galilei algebra that is algebra of invariance for two well-known linear equations: heat equation and Schroedinger equation. It turns out that there are no any nonlinear heat equations preserving this Galilei algebra, however, a wide class of nonlinear Schroedinger type equations with Galilei invariance exists. We show that the similar results are obtained for so called Schroedinger algebra that is the most popular extension of Galilei algebra. The results obtained are illustrated by the examples of the nonlinear Schroedinger equations, the Hamilton-Jacobi type systems and reaction-diffusion equations, which arise in applications. Further we study the conformal Galilei algebra and so called exotic conformal Galilei algebra. This is shown that there are no single second-order PDEs invariant under the conformal Galilei algebra but systems of PDEs only can admit this algebra. Finally a representation of exotic conformal Galilei algebra that obtained very recently is studied. Systems of PDEs, which are invariant under this algebra, are constructed. An example of a such system is presented and shown that one might be of interest in applications, for instance in magnetohydrodynamics.