Smooth multisoliton solutions and their peakon limit of Novikov's Camassa-Holm type equation with cubic nonlinearity

TL;DR

This paper presents explicit multisoliton solutions for Novikov's Camassa-Holm type equation, explores their structure and limits to peakons, and reveals connections to shallow-water wave models and conservation laws.

Contribution

It provides a compact parametric form of multisoliton solutions, analyzes their peakon limits, and establishes links to related shallow-water wave equations and conservation laws.

Findings

Smooth multisoliton solutions are explicitly constructed.

Solutions converge to peakons as background tends to zero.

Phase shifts of solitons match those of the Degasperis-Procesi equation.

Abstract

We consider Novikov's Camassa-Holm type equation with cubic nonlinearity. In particular, we present a compact parametric representation of the smooth bright multisolution solutions on a constant background and investigate their structure. We find that the tau-functions associated with the solutions are closely related to those of a model equation for shallow-water waves (SWW) introduced by Hirota and Satsuma. This novel feature is established by applying the reciprocal transformation to the Novikov equation. We also show by specifying a complex phase parameter that the smooth soliton is converted to a novel singular soliton with single cusp and double peaks. We demonstrate that both the smooth and singular solitons converge to a peakon as the background field tends to zero whereby we employ a method that has been developed for performing the similar limiting procedure for the multisoliton solutions of the Camassa-Holm equation. In the subsequent asymptotic analysis of the two- and $N$-soliton solutions, we confirm their solitonic behaviour. Remarkably, the formulas for the phase shifts of solitons as well as their peakon limits coincide formally with those of the Degasperis-Procesi equation. Last, we derive an infinite number of conservation laws of the Novikov equation by using a relation between solutions of the Novikov equation and those of the SWW equation. In appendix, we prove various bilinear identities associated with the tau-functions of the multisoliton solutions of the SWW equation.