On solutions of the Camassa-Holm equation. (English) Zbl 1039.76006

Johnson, R. S.

Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 459, No. 2035, 1687-1708 (2003).

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:

76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q53	KdV equations (Korteweg-de Vries equations)
35-04	Software, source code, etc. for problems pertaining to partial differential equations
35K40	Second-order parabolic systems
35P25	Scattering theory for PDEs
35Q51	Soliton equations
35R30	Inverse problems for PDEs
76B25	Solitary waves for incompressible inviscid fluids

Keywords:

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

Software:

Maple; Mathematica