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 (κ→0) in which the cuspon converges to an antipeakon. 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.