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Traveling wave solutions of the Camassa-Holm equation. (English) Zbl 1082.35127

The Camassa-Holm equation (CHE) arises as a model for the unidirectional propagation of shallow water waves over a flat bottom, as well as a model for nonlinear waves in a cylindrical axially symmetric hyperelastic rod. CHE is a bi-Hamiltonian equation with infinitely many conservation laws. The author studies traveling wave solutions of the CHE using a natural weak formulation. It is shown that, in addition to smooth solutions, there are a multitude of traveling waves with singularities: peakons, cuspons, stumpons, and composite waves.

MSC:

35Q35 PDEs in connection with fluid mechanics
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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