Communications in Mathematical Sciences

Volume 16 (2018)

Number 5

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

Pages: 1361 – 1378

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n5.a9

Authors

Yue Pu (Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

Robert L. Pego (Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

Denys Dutykh (Université Grenoble Alpes, Université Savoie Mont Blanc, Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques, Chambéry, France)

Didier Clamond (Université Côte d’Azur, Laboratoire J.A. Dieudonné, Centre National de la Recherche Scientifique (CNRS), Nice, France)

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., 55:237–247, 2018]. 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.

Keywords

Serre equations, Green–Naghdi equations, shallow water, weak solutions, long waves, peakons, cuspons, energy loss

2010 Mathematics Subject Classification

35L67, 35Q35, 76B15, 76B25, 76M22

Full Text (PDF format)

Received 20 February 2018

Received revised 10 May 2018

Accepted 10 May 2018

Published 19 December 2018