An integrable evolution equation for surface waves in deep water

TL;DR

This paper derives a new integrable nonlinear model for deep water surface waves, capturing their dynamics and breaking behavior, with theoretical and numerical validation showing its effectiveness.

Contribution

It introduces a novel integrable evolution equation for deep water waves, including its Lax pair, conserved quantities, and symmetry properties.

Findings

Supports periodic Stokes waves that peak and break in finite time

The model is completely integrable and exhibits soliton solutions

Comparison with classical results validates the model's accuracy

Abstract

In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative methods, an asymptotic model for small-aspect-ratio waves is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical irrotational results is performed.