Spatial Form of a Hamiltonian Dysthe Equation for Deep-Water Gravity Waves

Philippe Guyenne,Adilbek Kairzhan,Boyang Xu,Catherine Sulem

doi:10.3390/fluids6030103

Philippe Guyenne, Adilbek Kairzhan + Show 2 more

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https://doi.org/10.3390/fluids6030103

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Abstract

An overview of a Hamiltonian framework for the description of nonlinear modulation of surface water waves is presented. The main result is the derivation of a Hamiltonian version of Dysthe’s equation for two-dimensional gravity waves on deep water. The reduced problem is obtained via a Birkhoff normal form transformation which not only helps eliminate all non-resonant cubic terms but also yields a non-perturbative procedure for surface reconstruction. The free surface is reconstructed from the wave envelope by solving an inviscid Burgers’ equation with an initial condition given by the modulational Ansatz. Particular attention is paid to the spatial form of this model, which is simulated numerically and tested against laboratory experiments on periodic groups and short-wave packets. Satisfactory agreement is found in all these cases.

Highlights

Because the cubic nonlinear Schrödinger (NLS) equation for water waves had been shown to have a limited range of applicability, Dysthe [1] extended this model to the order in the perturbation analysis
We focus on the spatial version (28) and test it against laboratory experiments by Keller [42] on periodic groups and by Su [43] on short-wave packets
We have given an overview of the Hamiltonian modulational approach that has been advocated in a series of papers by Craig et al [21,22,23,24,25] and which leads to a Hamiltonian version of Dysthe’s equation for two-dimensional gravity waves on deep water, as recently shown in [20]

Summary

Introduction

Due to its higher-order nonlinear terms, Dysthe’s equation is able to capture the asymmetric development of propagating wave packets. Derived in the context of gravity waves on deep water, it has since been extended to various other settings and has produced a large literature: e.g., finite depth [2], gravitycapillary waves [3], exact linear dispersion [4], dissipation [5], higher order [6], just to cite a few references. Interest in Dysthe’s equation lies in the fact that it is more amenable to mathematical analysis and numerical simulation than the full water wave equations, while being able to capture salient features of weakly nonlinear wave packets. The associated computational cost is significantly less compared to that from direct (fully nonlinear) numerical solvers [7,8,9,10]

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Conclusion