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Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system. (English) Zbl 1231.35184
Summary: Considered herein is a two-component Camassa-Holm system modeling shallow water waves moving over a linear shear flow. It is shown here that solitary-wave solutions of the system are dynamically stable to perturbations for a range of their speeds. On the other hand, a new wave-breaking criterion for solutions is established, and two results of wave-breaking solutions with certain initial profiles are described in detail. Moreover, a sufficient condition for global solutions determined only by a nonzero initial profile of the free surface component of the system is found.
MSC:
35Q35PDEs in connection with fluid mechanics
35B35Stability of solutions of PDE
35C08Soliton solutions of PDE
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction