Two kinds of peaked solitary waves of the KdV, BBM and Boussinesq equations

TL;DR

This paper presents the first closed-form solutions of peaked solitary waves for the KdV, BBM, and Boussinesq equations, suggesting such waves may be common in shallow water models.

Contribution

It introduces novel peaked solitary wave solutions for several classical shallow water equations, expanding understanding of wave phenomena beyond the Camassa-Holm model.

Findings

Peaked solitary waves exist for KdV, BBM, and Boussinesq equations.

All solutions satisfy Rankine-Hugoniot jump conditions as weak solutions.

Peaked waves may be prevalent in various shallow water models.

Abstract

It is well-known that the celebrated Camassa-Holm equation has the peaked solitary waves, which have been not reported for other mainstream models of shallow water waves. In this letter, the closed-form solutions of peaked solitary waves of the KdV equation, the BBM equation and the Boussinesq equation are given for the first time. All of them have either a peakon or an anti-peakon. Each of them exactly satisfies the corresponding Rankine-Hogoniot jump condition and should be understood as weak solution. Therefore, the peaked solitary waves might be common for most of shallow water wave models, no matter whether or not they are integrable and/or admit breaking-wave solutions.