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arXiv · 1204.2889

Special solutions to a compact equation for deep-water gravity waves

Francesco Fedele·Denys Dutykh
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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.

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physics.class-phcs.NAmath.APmath.NAphysics.ao-phphysics.comp-phphysics.flu-dyn

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BibTeXRIS

Francesco Fedele, Denys Dutykh. 2012-04-13. Special solutions to a compact equation for deep-water gravity waves. https://doi.org/10.1017/jfm.2012.447

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