Reduction of Hugoniot–Maslov chains for trajectories of solitary vortices of the “shallow water” equations to the Hill equation

According to V. Maslov's idea, many of 2D quasilinear hyperbolic systems of PDE posses only 3 types of singularities in generic positions with properties of “structure” self-similarity and stability. They are shock waves, “narrow” solitons and “square root” point singularities (solitary vortices). Their propogations are described by infinite chains of ODE that generalize the well known Hugoniot conditions for shock waves. After some resonable closing of the chain for solitary vortices of the “shallow water” equations we obtain the nonlinear system of 16 ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. It means that in some approximation the trajectory of solitary vortex can be decribed by the Hill equation. This result can be used also for prediction of a future trajectory of the centre of solitary vortices via its observable part.