Hugoniót–Maslov Chains for Singular Vortical Solutions to Quasilinear Hyperbolic Systems and Typhoon Trajectory

Abstract

According to Maslov, many 2D quasilinear systems of PDE possess only three algebras of singular solutions with properties of “structural” self-similarity and stability. They are the algebras of shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by infinite chains of ODE (the Hugoniót–Maslov chains). We consider the Hugoniót-Maslov chain for the “square-root” point singularities of the shallow water equations. We discuss different related mathematical questions (in particular, unexpected integrability effects) as well as their possible application to the problem of typhoon dynamics.