Nonlinear resonances of water waves
arXiv:0905.0050 · doi:10.3934/dcdsb.2009.12.607
Abstract
In the last fifteen years, a great progress has been made in the understanding of the 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 have been discovered, conception of a resonance cluster has been much and successful 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 -class method which is effective for computing integer points on the resonance manifolds, and b) construction of the 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 formulate three most important open problems at the end.
14 pages, 3 figures, to appear in DCDS, final version (small changes in the text, type errors corrected, some additional bibliographic items added)
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Cited by in corpus (4)
- On the time-evolution of resonant triads in rotational capillary-gravity water waves
- On the modelling of short and intermediate water waves
- The Lagrangian formulation for wave motion with a shear current and surface tension
- A constructive method for computing generalized Manley--Rowe constants of motion