Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Nonlinear resonances of water waves

Pages: 607 - 621, Volume 12, Issue 3, October 2009      doi:10.3934/dcdsb.2009.12.607

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Elena Kartashova - RISC, J. Kepler University, Altenbergerstr. 69, Linz, A-4040, Austria (email)

Abstract: In the last fifteen years great progress has been made in the understanding of nonlinear resonance dynamics of water waves. Notions of scale- and angle-resonances have been introduced, new type of energy cascade due to nonlinear resonances in the gravity water waves has been discovered, conception of a resonance cluster has been much and successfully employed, a novel model of laminated wave turbulence has been developed, etc. etc. Two milestones in this area of research have to be mentioned: a) development of the $q$-class method which is effective for computing integer points on resonance manifolds, and b) construction of marked planar graphs, instead of classical resonance curves, representing simultaneously all resonance clusters in a finite spectral domain, together with their dynamical systems. Among them, new integrable dynamical systems have been found that can be used for explaining numerical and laboratory results. The aim of this paper is to give a brief overview of our current knowledge about nonlinear resonances among water waves, and finally to formulate the three most important open problems.

Keywords:  Nonlinear resonances, dynamical systems, fluid mechanics.
Mathematics Subject Classification:  Primary: 74J30, 37N10; Secondary: 37-02.

Received: April 2009;      Revised: June 2009;      Available Online: July 2009.