Hugoniot-Maslov chains for solitary vortices of the shallow water equations, I. - Derivation of the chains for the case of variable Coriolis forces and reduction to the Hill equation

The dynamics of solitary weak point singularities (vortices) is studied for the system of shallow water equations with variable Coriolis force. Accord ing to Maslov's viewpoint, this dynamics can be described by an infinite ch ain of ordinary differential equations similar to the Hugoniot conditions a rising in the theory of shock waves. In the first part we show that a physi cally reasonable truncation of the chain results in a system of sixteen non linear ordinary differential equations which is equivalent to a family of H ill equations. In other words, in some approximation the vortex in question can be treated as a rigid body whose trajectory is determined by the Hill equation. Part II (to be published later) deals with the study of various t rajectories of vortices, the influence of the Coriolis force depending on l atitude tin the beta-plane approximation), and the relationship between the behavior of the trajectories and the characteristics of solutions of the H ill equation. Applications to the problem on the typhoon eye trajectory wil l also be considered.