Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation

TL;DR

This paper derives explicit multipeakon solutions for Novikov's integrable cubic nonlinear Camassa-Holm type equation using spectral methods and matrix Lax pairs, expanding understanding of peakon solutions in nonlinear integrable systems.

Contribution

It provides the first explicit formulas for multipeakon solutions of Novikov's cubic Camassa-Holm equation, linking spectral problems to cubic string equations.

Findings

Explicit multipeakon solutions derived

Spectral problem related to cubic string equation

Connections made to previous work on Degasperis-Procesi equation

Abstract

Recently Vladimir Novikov found a new integrable analogue of the Camassa-Holm equation, admitting peaked soliton (peakon) solutions, which has nonlinear terms that are cubic, rather than quadratic. In this paper, the explicit formulas for multipeakon solutions of Novikov's cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang. By a transformation of Liouville type, the associated spectral problem is related to a cubic string equation, which is dual to the cubic string that was previously found in the work of Lundmark and Szmigielski on the multipeakons of the Degasperis-Procesi equation.