The peakon limits of soliton solutions of the Camassa-Holm equation

A method for obtaining peakon limits of multisoliton solutions of the Camassa-Holm equation is proposed and used to recover the peakon and two-peakon limits of the solitary wave and two-soliton solution, respectively. The limiting procedure is based on a novel representation of the soliton solutions-called PQ-decomposition-that is introduced in the study. The results shed light on the interaction dynamics of the two-soliton: it is shown that any single-crested collision eventually breaks down into a double-humped soliton as we proceed to the peakon limit. A criterion is obtained that discriminates between this dynamical behaviour of the two-soliton solutions and, by extension, determines the breakdown point in the interaction. This can be viewed as a direct analogue of the classical result for the Korteweg-de Vries equation whereby a critical amplitude-ratio dictates whether a two-soliton forms a single- or double-peaked wave at collision.