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

Published online by Cambridge University Press:  16 October 2012

Francesco Fedele  and
Denys Dutykh
Francesco Fedele*
Affiliation:
School of Civil and Environmental Engineering & School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
Denys Dutykh
Affiliation:
LAMA UMR 5127 CNRS, Université de Savoie, Campus Scientifique, 73376 Le Bourget-du-Lac CEDEX, France
*
†Email address for correspondence: [email protected]
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Abstract

Dyachenko & Zakharov (J. Expl Theor. Phys. Lett., vol. 93, 2011, pp. 701–705) recently 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 travelling wave solutions of this compact equation are numerically constructed using the Petviashvili method. Their stability properties are also investigated. In particular, unstable travelling waves with wedge-type singularities, namely peakons, are numerically discovered. To gain insight into the properties of these singular solutions, we also 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.

JFM classification

Waves/Free-surface Flows: Solitary waves Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 712 , 10 December 2012 , pp. 646 - 660
Copyright
©2012 Cambridge University Press

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