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.
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Francesco Fedele, Denys Dutykh. Special solutions to a compact equation for deep-water gravity waves. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2012, 712, pp.646-660. ⟨http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8764936&fulltextType=RA&fileId=S0022112012004478⟩. ⟨10.1017/jfm.2012.447⟩. ⟨hal-00687325v3⟩



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