Cusped solitons of the Camassa-Holm equation. II. Binarycuspon-soliton interactions

摘要

This paper extends the results of a previous work [Parker A. Cusped solitons of the Camas-sa-Holm equation. I. Cuspon solitary wave and antipeakon limit. Chaos, Solitons & Fractals2007;34:730-91-designated I-in which the solitary cuspon solution of the Camassa-Holm equation and its antipeakon limit were considered. Here, explicit binary cuspon-soliton solutions are obtained in parametric form by exploiting the factorisation procedurethat was used in I. The structure and dynamics of these two-wave interactions are inves-tigated and some unanswered questions concerning the waveforms are addressed. In par-ticular, it is shown that, while a cuspon may be 'swallowed-up' by a soliton, the converse isnever possible regardless of the depth of the cuspon trough. Formulae for the characteristicpost-collision phase shifts are obtained and analysed in detail; this permits a reassessmentof the previously reported results. Examples of twin-cusped and mixed cuspon-solitonsolutions are presented within the different parameter regimes. Coincidentally, we findnew two-soliton solutions of the related associated-Camassa-Holm equation. Thesedescribe binary interactions of classically smooth elevated solitons and solitary-wavetroughs.