Relative-Periodic Elastic Collisions of Water Waves

TL;DR

This paper constructs and analyzes time- and relative-periodic solutions of the free-surface Euler equations, demonstrating elastic collisions of water waves that resemble soliton interactions, extending beyond integrable models.

Contribution

It introduces a method to find exact periodic solutions of water waves exhibiting elastic collisions, showing such phenomena occur outside integrable regimes.

Findings

Solutions can be made radiation-free, behaving like solitons.

Larger waves can subsume smaller ones upon collision.

Elastic collisions occur outside the KdV regime.

Abstract

We compute time-periodic and relative-periodic solutions of the free-surface Euler equations that take the form of overtaking collisions of unidirectional solitary waves of different amplitude on a periodic domain. As a starting guess, we superpose two Stokes waves offset by half the spatial period. Using an overdetermined shooting method, the background radiation generated by collisions of the Stokes waves is tuned to be identical before and after each collision. In some cases, the radiation is effectively eliminated in this procedure, yielding smooth soliton-like solutions that interact elastically forever. We find examples in which the larger wave subsumes the smaller wave each time they collide, and others in which the trailing wave bumps into the leading wave, transferring energy without fully merging. Similarities notwithstanding, these solutions are found quantitatively to lie outside of the Korteweg-de Vries regime. We conclude that quasi-periodic elastic collisions are not unique to integrable model water wave equations when the domain is periodic.