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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Non-persistence of roll-waves under viscous perturbations

Pages: 61 - 70, Volume 1, Issue 1, February 2001      doi:10.3934/dcdsb.2001.1.61

 
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Pascal Noble - Mathématiques pour I'Industrie et la Physique, UFR MIG 118, route de Narbonne, 31062 Toulouse Cedex, France (email)
Sebastien Travadel - Mathématiques pour I'Industrie et la Physique, UFR MIG 118, route de Narbonne, 31062 Toulouse Cedex, France (email)

Abstract: In this paper, we study the existence of periodic traveling waves for the equations of the Shallow-water theory with a small viscosity. This small viscosity leads to a singularly perturbed problem. The slow-fast system involves a point where normal hyperbolicity breaks down. We first prove the existence of a slow manifold around this point. Then we show that there are no periodic solutions for small viscosity and describe completely the structure of a travelling wave.

Keywords:  Shallow-water equations, roll waves, singular perturbations.
Mathematics Subject Classification:  34C25.

Received: October 2000;      Revised: January 2001;      Published: January 2001.