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On solutions of the Camassa-Holm equation. (English) Zbl 1039.76006
Summary: The techniques that have been developed for the application of the inverse-scattering transform method to the solution of Camassa-Holm equation, principally by Constantin, are implemented. We use this approach, first, to represent the known solitary-wave solution in a simple parametric form, and then, second, to obtain the general two- and three-soliton solutions. (These latter two solutions require rather extensive use of mathematical packages, Mathematica and Maple, in order to complete the construction of solutions.) A number of examples are presented; the phase shifts, evident after an interaction, are found, and the special limit that recovers the peakon solutions is discussed.
MSC:
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q53KdV-like (Korteweg-de Vries) equations
35-04Machine computation, programs (partial differential equations)
35K40Systems of second-order parabolic equations, general
35P25Scattering theory (PDE)
35Q51Soliton-like equations
35R30Inverse problems for PDE
76B25Solitary waves (inviscid fluids)
Software:
Maple