On weakly singular and fully nonlinear travelling shallow capillary-gravity waves in the critical regime

TL;DR

This paper analyzes fully nonlinear, weakly dispersive models of capillary-gravity waves, revealing stable peaked solitary waves in the critical regime where surface tension balances gravity, with implications for wave dynamics.

Contribution

It introduces stable peaked solitary wave solutions in a fully nonlinear model at the critical Bond number, highlighting their properties and dynamics.

Findings

Existence of stable peaked solitary waves at the critical regime.

Wave solutions satisfy the classical speed-amplitude relation.

Decay rate of solutions is independent of amplitude.

Abstract

In this Letter we consider long capillary-gravity waves described by a fully nonlinear weakly dispersive model. First, using the phase space analysis methods we describe all possible types of localized travelling waves. Then, we especially focus on the critical regime, where the surface tension is exactly balanced by the gravity force. We show that our long wave model with a critical Bond number admits stable travelling wave solutions with a singular crest. These solutions are usually referred to in the literature as peakons or peaked solitary waves. They satisfy the usual speed-amplitude relation, which coincides with Scott-Russel's empirical formula for solitary waves, while their decay rate is the same regardless their amplitude. Moreover, they can be of depression or elevation type independent of their speed. The dynamics of these solutions are studied as well.