Hamiltonian description and traveling waves of the spatial Dysthe equations

TL;DR

This paper uncovers the Hamiltonian structure of the spatial Dysthe equations, develops a spectral scheme for their solution, and investigates traveling waves, revealing inelastic collisions and non-integrability.

Contribution

It reveals the Hamiltonian structure and invariants of the spatial Dysthe equations, introduces a spectral scheme for their numerical solution, and studies traveling wave collisions.

Findings

Conservation laws are satisfied up to machine precision.

Traveling wave collisions are inelastic.

The equations are likely non-integrable.

Abstract

The spatial version of the fourth-order Dysthe equations describe the evolution of weakly nonlinear narrowband wave trains in deep waters. For unidirectional waves, the hidden Hamiltonian structure and new invariants are unveiled by means of a gauge transformation to a new canonical form of the evolution equations. A highly accurate Fourier-type spectral scheme is developed to solve for the equations and validate the new conservation laws, which are satisfied up to machine precision. Further, traveling waves are numerically investigated using the Petviashvili method. It is found that their collision appears inelastic, suggesting the non-integrability of the Dysthe equations.