On the Camassa-Holm equation and a direct method of solution. II. Soliton solutions

Previous attempts to find explicit analytic multisoliton solutions of the general Camassa-Holm (CH) equation have met with limited success. This study (which falls into two parts, designated II and III) extends the results of the prior work (1) in which a bilinear form of the CH equation was constructed and then solved for the solitary-wave solutions. It is shown that Hirota's bilinear transformation method can be used to derive exact multisoliton solutions of the equation in a systematic way. Here, analytic two-soliton solutions are obtained explicitly and their structure and dynamics are investigated in the different parameter regimes, including the limiting 'two-peakon' form. The solutions possess a non-standard representation that is characterized by an additional parameter, and the structure of this key parameter is examined. These results pave the way for constructing the hallmark N-soliton solutions of the CH equation in part III.