On solutions of the Camassa-Holm equation

R. S. Johnson

Abstract

The techniques that have been developed for the application of the inverse–scattering transform method to the solution of the 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.