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The magnetohydrodynamic equations system for heavy fluid over an arbitrarysurface in shallow water approximation is studied in the present paper. It isshown that simple wave solutions exist only for underlying surfaces that areslopes of constant inclination. All self-similar discontinuous and continuoussolutions are found. The exact explicit solutions of initial discontinuitydecay problem over a flat plane and a slope are found. It is shown that theinitial discontinuity decay solution is represented by one of five possiblewave configurations. For each configuration the necessary and sufficientconditions for its realization are found. The change of dependent andindependent variables transforming the initial equations over a slope to thoseover a flat plane is found.

Nonlinear dynamics of magnetohydrodynamic flows of heavy fluid over an arbitrary surface in shallow water approximation