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Explicit solutions of the Camassa-Holm equation. (English) Zbl 1072.35156
Summary: Explicit travelling-wave solutions of the Camassa-Holm equation are found. The solutions are characterized by two parameters. For propagation in the positive \(x\)-direction, both periodic and solitary smooth-hump, peakon, cuspon and inverted-cuspon waves are found. For propagation in the negative \(x\)-direction, there are solutions which are just the mirror image in the \(x\)-axis of the aforementioned solutions. Some composite wave solutions of the Degasperis-Procesi equation are given in an appendix.

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