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Zbl 1039.76006
Johnson, R.S.
On solutions of the Camassa-Holm equation. (English)
[J] Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 459, No. 2035, 1687-1708 (2003). ISSN 1364-5021; ISSN 1471-2946/e

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 2000:: *76B15 Wave motions (fluid mechanics)
35Q53 KdV-like equations
35-04 Machine computation, programs (partial differential equations)
35K40 Systems of parabolic equations, general
35P25 Scattering theory (PDE)
35Q51 Solitons
35R30 Inverse problems for PDE
76B25 Solitary waves, etc. (inviscid fluids)

Keywords: three-soliton solutions; peakons; inverse scattering methods; Korteweg-de Vries equation

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Zentralblatt MATH Berlin [Germany]