Hamiltonian Dysthe equation for 3D deep-water gravity waves

Philippe Guyenne; Adilbek Kairzhan; Catherine Sulem

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arXiv · 2108.10822

Hamiltonian Dysthe equation for 3D deep-water gravity waves

Philippe Guyenne·Adilbek Kairzhan·Catherine Sulem

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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.

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Philippe Guyenne, Adilbek Kairzhan, Catherine Sulem. 2021-08-24. Hamiltonian Dysthe equation for 3D deep-water gravity waves. https://arxiv.org/abs/2108.10822

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