Singular solutions for the shallow-water equations

Abstract

The method of weak asymptotics is used to find singular solutions of the shallow-water system which can contain Dirac-δ distributions (Espinosa & Omel'yanov, 2005). Complex-valued approximations which become real-valued in the distributional limit are shown to extend the range of possible singular solutions. It is shown, in this paper, how this approach can be used to construct solutions containing combinations of classical hyperbolic shock waves and Dirac-δ distributions. Uniqueness is obtained in a smaller class of distributions which satisfies a condition of Oleinik type and minimizes the number of δ-singularities.