Hamiltonian form and solitary waves of the spatial Dysthe equations

F. Fedele; D. Dutykh

doi:10.1134/S0021364011240039

Hamiltonian form and solitary waves of the spatial Dysthe equations

F. Fedele
, D. Dutykh

Mathematics

School of Electrical and Computer Engineering

Research output: Contribution to journal › Article › peer-review

17 Scopus citations

Abstract

The spatial Dysthe equations describe the envelope evolution of the free-surface and potential of gravity waves in deep waters. Their Hamiltonian structure and new invariants are unveiled by means of a gauge transformation to a new canonical form of the evolution equations. An accurate Fourier-type spectral scheme is used to solve for the wave dynamics and validate the new conservation laws, which are satisfied up to machine precision. Further, traveling waves are numerically constructed using the Petviashvili method. It is shown that their collision appears inelastic, suggesting the non-integrability of the Dysthe equations.

Original language	British English
Pages (from-to)	840-844
Number of pages	5
Journal	JETP Letters
Volume	94
Issue number	12
DOIs	https://doi.org/10.1134/S0021364011240039
State	Published - Feb 2012

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title = "Hamiltonian form and solitary waves of the spatial Dysthe equations",

abstract = "The spatial Dysthe equations describe the envelope evolution of the free-surface and potential of gravity waves in deep waters. Their Hamiltonian structure and new invariants are unveiled by means of a gauge transformation to a new canonical form of the evolution equations. An accurate Fourier-type spectral scheme is used to solve for the wave dynamics and validate the new conservation laws, which are satisfied up to machine precision. Further, traveling waves are numerically constructed using the Petviashvili method. It is shown that their collision appears inelastic, suggesting the non-integrability of the Dysthe equations.",

author = "F. Fedele and D. Dutykh",

note = "Funding Information: D. Dutykh acknowledges the support of French Agence Nationale de la Recherche, project Math Ocean (Grant ANR 08 BLAN 0301 01). F. Fedele acknowledges the travel support received by the Geo physical Fluid Dynamics (GFD) Program to attend part of the summer school on “Shear Turbulence: Onset and Structure” at the Woods Hole Oceano graphic Institution in August 2011. We are grateful to Profs. Didier Clamond, Taras Lakoba, Paul Milewski, Lev Shemer, and Jianke Yang for useful discussions on the subject of nonlinear waves. F. Fedele also thanks Professor Phil Morrison for useful discussions and a short lecture on Hamiltonian systems during the 2011 GFD program.",

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doi = "10.1134/S0021364011240039",

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AU - Dutykh, D.

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N2 - The spatial Dysthe equations describe the envelope evolution of the free-surface and potential of gravity waves in deep waters. Their Hamiltonian structure and new invariants are unveiled by means of a gauge transformation to a new canonical form of the evolution equations. An accurate Fourier-type spectral scheme is used to solve for the wave dynamics and validate the new conservation laws, which are satisfied up to machine precision. Further, traveling waves are numerically constructed using the Petviashvili method. It is shown that their collision appears inelastic, suggesting the non-integrability of the Dysthe equations.

AB - The spatial Dysthe equations describe the envelope evolution of the free-surface and potential of gravity waves in deep waters. Their Hamiltonian structure and new invariants are unveiled by means of a gauge transformation to a new canonical form of the evolution equations. An accurate Fourier-type spectral scheme is used to solve for the wave dynamics and validate the new conservation laws, which are satisfied up to machine precision. Further, traveling waves are numerically constructed using the Petviashvili method. It is shown that their collision appears inelastic, suggesting the non-integrability of the Dysthe equations.

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DO - 10.1134/S0021364011240039

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VL - 94

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JO - JETP Letters

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IS - 12

ER -

Hamiltonian form and solitary waves of the spatial Dysthe equations

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