Special solutions to a compact equation for deep-water gravity waves
Abstract
Recently, Dyachenko & Zakharov (2011) have derived a compact form of the well known Zakharov integro-differential equation for the third order Hamiltonian dynamics of a potential flow of an incompressible, infinitely deep fluid with a free surface. Special traveling wave solutions of this compact equation are numerically constructed using the Petviashvili method. Their stability properties are also investigated. Further, unstable traveling waves with wedge-type singularities, viz. peakons, are numerically discovered. To gain insights into the properties of singular traveling waves, we consider the academic case of a perturbed version of the compact equation, for which analytical peakons with exponential shape are derived. Finally, by means of an accurate Fourier-type spectral scheme it is found that smooth solitary waves appear to collide elastically, suggesting the integrability of the Zakharov equation.
Cite
@article{arxiv.1204.2889,
title = {Special solutions to a compact equation for deep-water gravity waves},
author = {Francesco Fedele and Denys Dutykh},
journal= {arXiv preprint arXiv:1204.2889},
year = {2020}
}
Comments
17 pages, 14 figures, 41 references. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh/