Nonlinear dynamics of magnetohydrodynamic shallow water flows over an arbitrary surface

Abstract

The magnetohydrodynamic equations system for a heavy fluid over an arbitrary surface in shallow water approximation is studied. All self-similar discontinuous and continuous solutions are found. The exact explicit solutions of the initial discontinuity decay problem over a flat plane and a slope are found. It is shown that the initial discontinuity decay solution is represented by one of five possible wave configurations. For each configuration the necessary and sufficient conditions for its realization are found. The changes in dependent and independent variables transforming the initial equations over a slope into those over a flat plane are found.