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

• Main Authors: Philippe Guyenne, Adilbek Kairzhan, Catherine Sulem, Boyang Xu
• Format: Article
• Language: English
• Published: MDPI AG 2021-03-01
• Series: Fluids
• Subjects: deep-water gravity waves; Dysthe equation; Hamiltonian systems; modulation theory; numerical simulations
• Online Access: https://www.mdpi.com/2311-5521/6/3/103

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.