Nonlinear resonances of water waves

TL;DR

This paper reviews recent advances in understanding nonlinear water wave resonances, including new models, computational methods, and the identification of integrable systems, highlighting key open problems in the field.

Contribution

It provides a comprehensive overview of recent developments such as the q-class method, resonance clusters, and new integrable models for water wave nonlinear resonances.

Findings

Development of the q-class method for resonance computations

Introduction of resonance clusters and their dynamical systems

Discovery of new integrable dynamical systems for water waves

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 $q$-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.