Travelling breaking waves

We study a mathematical model of coastal waves in the shallow water approximation. The model contains two empirical parameters. The first one controls turbulent dissipation. The second one is responsible for the turbulent viscosity and is determined by the turbulent Reynolds number. We study travelling waves solutions to this model. The existence of an analytical and numerical solution to the problem in the form of a traveling wave is shown. The singular points of the system are described. It is shown that there exists a critical value of the Reylnols number corresponding to the transition from a monotonic profile to an oscillatory one. The paper is organized as follows. First, we present the governing system of ordinary differential equations (ODE) for travelling waves. Second, the Lyapunov function for the corresponding ODE system is derived. Finally, the behavior of the solution to the ODE system is discussed.