Alexander Chupakhin

We investigate the partial invariant solution of the system of the equations of the magneto hydrodynamics (MHD). This solution describes plane, steady motions of infinitely conducting gas in attendance of a magnetic field. The key-equation is the Bendikson equation type with degenerated singular point. We research topology of integral curves in a neighborhood of this singular point and infinity applying method of Frommer. There are two regimes of gas motions.

We investigate exact shallow water on a rotating sphere. Thismodel is used in oceanology and physics of atmosphere for describing large-scalemotions of gas and fluid.We construct and study solution,which describe the damped ring source on the sphere. The motion takes place in a spherical belt. System of equations of shallow water on the sphere has solutions of two types: supercritical (supersonic) and subcritical (subsonic).

Nonlinear Schrodinger equation (NSE) has many applications in mathematical physics (nonlinear optics, wave theory and so on). Gagnon and Winternitz have constructed symmetry algebra $L_{12}$ and optimal system of subalgebras for NSE (1989). It’s an extension of Galilei algebra $L_{11}$ admitted gas dynamics equations. Its three-dimensional symmetry subalgebras generate 27 different submodels. List of all solutions corresponding to these algebras has been received in this paper. Most of this solutions have not investigate previously.