ar X iv : m at h - ph / 0 40 40 03 v 1 1 A pr 2 00 4 On Maslov Conjecture about Square Root Type Singular Solutions of the Shallow Water Equations ∗

About twenty years ago, V. P. Maslov [1] put forward the idea that numerous quasilinear hyperbolic systems have only finite number of singular solution in general position. These solutions are shock waves, “infinitely narrow” solitons and point singularities of the type of the square root of a quadratic form. He has also stated conjecture that such solutions for shallow water equation can describe the dynamics of mesoscale vortices in the atmosphere, and trajectories of singularities correspond to trajectories of these vortices. Interesting integrability properties of such solutions were found in works [2, 3, 4, 5], where the shallow water equations on the β-plane with variable Coriolis force [6, 7] were considered :