On the Camassa–Holm equation and a direct method of solution. III. N -soliton solutions

Abstract

In the concluding study (designated III), we modify the direct bilinear transformation method for solving the Camassa–Holm (CH) equation that was set down in part II of this work. We demonstrate its efficacy for finding analytic multisoliton solutions of the equation and give explicit expressions for the first few solitons. It is shown that, at each order N , the N -soliton has a non-standard representation that is characterized by an ‘extra’ parameter. The stucture of this parameter is investigated and a procedure for constructing the general N -soliton solution of the CH equation is presented.