Asymptotic solutions of the linear shallow-water equations with localized initial data

Abstract

A new asymptotic method for solving Cauchy problems with localized initial data (perturbation) for the linearized shallow-water equation is suggested. The solution is decomposed into two parts: waves and vortices. Metamorphosis of the profile takes place for the wave part: it is localized in the neighborhood of the initial point and later on it is localized in the neighborhood of the front (1-D curve on the plane). Initially, the front is a smooth curve, but as the time increases turning (focal) and self-intersection points might appear. The vortical part is localized in the neighborhood of a point moving along the trajectory of the basic velocity vector field. Both parts are described by very simple formulae taking into account the form of the initial perturbation. These formulae are expressed by means of elementary functions for some special choice of the initial data. Due to the profile-metamorphosis phenomenon the method leading to the final formulae is not elementary. It makes use of semiclassical approximations and ray expansions. It is based on the generalization of the construction known as the Maslov canonical operator and boundary-layer expansions.