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.
Domains
Fluid mechanics [physics.class-ph] Fluids mechanics [physics.class-ph] Numerical Analysis [math.NA] Analysis of PDEs [math.AP] Pattern Formation and Solitons [nlin.PS] Exactly Solvable and Integrable Systems [nlin.SI] Fluid Dynamics [physics.flu-dyn] Atmospheric and Oceanic Physics [physics.ao-ph] Computational Physics [physics.comp-ph]Origin | Files produced by the author(s) |
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