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VOLUME 94 | ISSUE 12 | PAGE 921
Hamiltonian form and solitary waves of the spatial Dysthe equations
F. Fedele+, D. Dutykh*
+School of Civil and Environmental Engineering and School of Electrical and Computer Engineering, Georgia Institute of Technology, Ga 30332-0355 Atlanta, USA
*LAMA, UMR 5127 CNRS, Universitéde Savoie, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France

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. Moreover, 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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