Wave Turbulence from an Integrable Shallow Water Model

Wave turbulence theory provides a statistical framework for describing energy transfer in weakly nonlinear dispersive wave systems. For surface gravity waves in deep water, the statistical description is given by Hasselmann's kinetic equation. However, in the case of one-dimensional (1-D) shallow water, the traditional understanding is that the physics is governed by integrable equations, such as the Korteweg-de Vries (KdV) equation, where wave kinetic theory has been considered trivial. In this study, we challenge the conventional view of 1-D shallow-water dynamics. We begin with the 1-D Kaup–Boussinesq (KB) system, which is another integrable system supporting counter-propagating waves, and show that its corresponding wave kinetic equation is non-trivial. This is followed by a numerical study of the KB system, demonstrating the spectral evolution toward stationary states, as well as the soliton dynamics that can emerge from this system. Our findings highlight the importance of a greater understanding of the relationship between integrability and the kinetic theory of nonlinear wave systems.