Hamiltonian form and solitary waves of the spatial Dysthe equations

TL;DR

This paper reveals the Hamiltonian structure and invariants of the spatial Dysthe equations, develops numerical schemes to validate conservation laws, and investigates solitary wave interactions, indicating non-integrability.

Contribution

It introduces a gauge transformation to a canonical form, uncovers new invariants, and numerically studies solitary wave collisions in the Dysthe equations.

Findings

Conservation laws are satisfied up to machine precision.

Traveling waves are constructed using the Petviashvili method.

Wave collisions are inelastic, indicating non-integrability.

Abstract

The spatial Dysthe equations describe the envelope evolution of the free-surface and potential of gravity waves in deep waters. Their Hamiltonian structure and new invariants are unveiled by means of a gauge transformation to a new canonical form of the evolution equations. An accurate Fourier-type spectral scheme is used to solve for the wave dynamics and validate the new conservation laws, which are satisfied up to machine precision. Traveling waves are numerically constructed using the Petviashvili method. It is shown that their collision appears inelastic, suggesting the non-integrability of the Dysthe equations.