Nonlinear Resonances of Water Waves

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. Notions of scaleand 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. 1. Exposition. 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. Let us go through all these points one by one, regarding for concreteness 2D-wavevectors. First, the simplest possible understanding of a (propagating) wave as a Fourier harmonics Ak exp i(k · x− ω t) (1) is obviously too simplified and does not include normal modes which are due to boundary conditions. Here x = (x1, x2 ) and time t are space and time variables correspondingly, ω = ω(k) is dispersion function and k is wavevector. For instance, 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] |Ak| sin ( π mx1 Lx1 ) sin ( π nx2 Lx2 ) sin ( β 2ω x1 + ωt ) . (2) 2000 Mathematics Subject Classification. Primary: 74J30, 37N10; Secondary: 37-02.