Cusped solitons of the Camassa-Holm equation. I. Cuspon solitary wave and antipeakon limit

A factorisaton method is used to obtain the cusped soliton of the Camassa-Holm equation in parametric form. It is shown how this piecewise analytic solution arises from an associated smooth solitary wave. The PQ-decomposition of the explicit solution is then used to determine the dispersionless limit (K -> 0) in which the cuspon converges to an anti-peakon. The special cuspon solution reported by Kraenkel and Zenchuk [Kraenkel RA, Zenchuk A. Camassa-Holm equation: transformation to deformed sinh-Gordon equations, cuspon and soliton solutions. J Phys A: Math Gen 1999;32:4733-47] is recovered and examined in the context of the parametric representation. The cusped solitary wave of a short-wave version of the Camassa-Holm model is also deduced from the cuspon in an appropriate limit. (C) 2007 Elsevier Ltd. All rights reserved.