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.
Highlights
In the last fifteen years great progress has been made in the understanding of nonlinear resonance dynamics of water waves which is the main subject of discrete wave turbulence
In this paper we will try to present a major part of known analytical, numerical and laboratory results on nonlinear resonances among water waves, in as strict mathematical language as possible. This is not a simple task due to the three-fold problem: 1) there is no strict definition of a wave; 2) there is no general agreement about the types of waves which should be called water waves; 3) the notions of resonance in physics and mathematics are different
Ak exp i(k · x − ω t) is obviously too simplified and does not include normal modes which are due to boundary conditions
Summary
In the last fifteen years great progress has been made in the understanding of nonlinear resonance dynamics of water waves which is the main subject of discrete wave turbulence. In this paper we will try to present a major part of known analytical, numerical and laboratory results on nonlinear resonances among water waves, in as strict mathematical language as possible. This is not a simple task due to the three-fold problem: 1) there is no strict definition of a wave; 2) there is no general agreement about the types of waves which should be called water waves; 3) the notions of resonance in physics and mathematics are different. The normal mode of oceanic planetary waves (that are due to the Earth rotation) with zero boundary conditions in a finite box [0, Lx1 ] × [0, Lx2 ] reads [44]. On we assume that dispersion function ω(k) defines the type of a wave, for instance, ω ∼ |k|3/2 corresponds to capillary waves in a rectangular box with periodic boundary conditions, ω ∼ |k|−1 corresponds to oceanic planetary waves in a rectangular box with zero boundary conditions, etc
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