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On an integrable two-component Camassa-Holm shallow water system. (English) Zbl 1227.76016
Summary: The interest in the Camassa-Holm equation inspired the search for various generalizations of this equation with interesting properties and applications. In this Letter we deal with such a two-component integrable system of coupled equations. First we derive the system in the context of shallow water theory. Then we show that while small initial data develop into global solutions, for some initial data wave breaking occurs. We also discuss the solitary wave solutions. Finally, we present an explicit construction for the peakon solutions in the short wave limit of system.
MSC:
76D33Waves in incompressible viscous fluids
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q30Stokes and Navier-Stokes equations
37J35Completely integrable systems, topological structure of phase space, integration methods
35C08Soliton solutions of PDE