Hamiltonian Dysthe equation for 3D deep-water gravity waves

TL;DR

This paper derives a Hamiltonian Dysthe equation for 3D deep-water gravity waves using normal form transformations, providing a refined model that closely matches numerical simulations and classical predictions.

Contribution

It introduces a new derivation of the Hamiltonian Dysthe equation for 3D waves, enhancing the understanding of wave envelope evolution in deep water.

Findings

The derived equation accurately predicts wave evolution in 3D deep water.

Numerical simulations confirm the model's high fidelity to the Euler system.

Refined reconstruction improves the description of the free surface.

Abstract

This article concerns the water wave problem in a three-dimensional domain of infinite depth and examines the modulational regime for weakly nonlinear wavetrains. We use the method of normal form transformations near the equilibrium state to provide a new derivation of the Hamiltonian Dysthe equation describing the slow evolution of the wave envelope. A precise calculation of the third-order normal form allows for a refined reconstruction of the free surface. We test our approximation against direct numerical simulations of the three-dimensional Euler system and against predictions from the classical Dysthe equation, and find very good agreement.