Cusp and loop soliton solutions of short-wave models for the Camassa–Holm and Degasperis–Procesi equations

Abstract A simple method is developed for constructing the solutions of the short-wave model equations associated with the Camassa–Holm (CH) and Degasperis–Procesi (DP) shallow-water wave equations. Taking an appropriate scaling limit of the N -soliton solution of the CH equation, we obtain the N -cusp soliton solution for the CH short-wave model. The similar procedure also leads to the N -loop soliton solution for the DP short-wave model. We describe the property of the solutions. In particular, we derive the large-time asymptotics of the solutions as well as the formulas for the phase shift.

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