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

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.
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Yue Pu, Robert Pego, Denys Dutykh, Didier Clamond. Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations. Communications in Mathematical Sciences, International Press, 2018, 16 (5), pp.1361-1378. ⟨10.4310/CMS.2018.v16.n5.a9⟩. ⟨hal-01794005⟩

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