Hamiltonian description and traveling waves of the spatial Dysthe equations
Abstract
The spatial version of the fourth-order Dysthe equations describe the evolution of weakly nonlinear narrowband wave trains in deep waters. For unidirectional waves, the hidden Hamiltonian structure and new invariants are unveiled by means of a gauge transformation to a new canonical form of the evolution equations. A highly accurate Fourier-type spectral scheme is developed to solve for the equations and validate the new conservation laws, which are satisfied up to machine precision. Further, traveling waves are numerically investigated using the Petviashvili method. It is found that their collision appears inelastic, suggesting the non-integrability of the Dysthe equations.
Keywords
Cite
@article{arxiv.1110.3605,
title = {Hamiltonian description and traveling waves of the spatial Dysthe equations},
author = {Francesco Fedele and Denys Dutykh},
journal= {arXiv preprint arXiv:1110.3605},
year = {2012}
}
Comments
Research report. 17 pages, 7 figures, 38 references. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh/ . arXiv admin note: substantial text overlap with arXiv:1110.4083