REDUCTION OF HUGONIOT-MASLOV CHAINS FOR TRAJECTORIES OF SOLITARY VORTICES OF THE SHALLOW-WATER EQUATIONS TO THE HILL EQUATION

According to Maslov's idea, many two-dimensional, quasilinear hyperbol ic systems of partial differential equations admit only three types of singularities that are in general position and have the property of ' 'structure self-similarity and stability.'' Those are: shock waves, '' narrow'' solitons, and ''square-root'' point singularities (solitary v ortices). Their propagation is described by an infinite chain of ordin ary differential equations (ODE) that generalize the well-known Hugoni ot conditions for shock waves. After some reasonable closure of the ch ain for the case of solitary vortices in the ''shallow water'' equatio ns, we obtain a nonlinear system of sixteen ODE, which is exactly equi valent to the (linear) Hill equation with a periodic potential. This m eans that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to pre dict the trajectory of the vortex center if we know its observable par t.