Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations

TL;DR

This paper investigates a non-dispersive regularization of shallow-water equations, revealing weak singular shock profiles, energy dissipation at shocks, and the existence of cusped solitary waves, contributing to understanding wave behavior in fluid dynamics.

Contribution

It introduces and analyzes weakly singular shock profiles and cusped solitary waves in a non-dispersive regularization of shallow-water equations, extending classical shock wave theory.

Findings

Existence of continuous, piecewise smooth shock solutions with weak singularities.

Energy dissipation occurs at the shock point.

Presence of cusped solitary wave solutions.

Abstract

We study a regularization of the classical Saint-Venant (shallow-water) equations, recently introduced by D. Clamond and D. Dutykh (Commun. Nonl. Sci. Numer. Simulat. 55 (2018) 237-247). This regularization is non-dispersive and formally conserves mass, momentum and energy. We show that for every classical shock wave, the system admits a corresponding non-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point where energy is dissipated as it is for the classical shock. The system also admits cusped solitary waves of both elevation and depression.