Normal Form Transformations and Dysthe’s Equation for the Nonlinear Modulation of Deep-Water Gravity Waves

A new Hamiltonian version of Dysthe’s equation is derived for two-dimensional weakly modulated gravity waves on deep water. A key ingredient in this derivation is a Birkhoff normal form transformation that eliminates all non-resonant cubic terms and allows for a refined reconstruction of the free su...

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Bibliographic Details
Published inWater Waves An interdisciplinary journal Vol. 3; no. 1; pp. 127 - 152
Main Authors Craig, Walter, Guyenne, Philippe, Sulem, Catherine
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.04.2021
Subjects
Mathematical Applications in the Physical Sciences
Mathematics
Mathematics and Statistics
Original Article
Partial Differential Equations
Hamiltonian systems
Normal forms
Dysthe’s equation
Modulation theory
Deep water
Dirichlet–Neumann operator
Gravity waves
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Summary:A new Hamiltonian version of Dysthe’s equation is derived for two-dimensional weakly modulated gravity waves on deep water. A key ingredient in this derivation is a Birkhoff normal form transformation that eliminates all non-resonant cubic terms and allows for a refined reconstruction of the free surface. This modulational approximation is tested against numerical solutions of the classical Dysthe’s equation and against direct numerical simulations of Euler’s equations for nonlinear water waves. Very good agreement is found in the context of Benjamin–Feir instability of Stokes waves, for which an analysis is provided. An extension of our Hamiltonian model incorporating exact linear dispersion as well as an alternate spatial form are also proposed.
ISSN:2523-367X
2523-3688
DOI:10.1007/s42286-020-00029-7