On the nonlinear dynamics of the traveling-wave solutions of the Serre system

TL;DR

This paper numerically investigates the nonlinear dynamics of traveling wave solutions in the Serre system, focusing on stability, interactions, and wave breaking, using high-order finite-element methods.

Contribution

It provides a detailed numerical analysis of the stability and interactions of traveling waves in the Serre equations, including comparisons with Euler equations.

Findings

Traveling waves exhibit stability and persistence under certain conditions.

Interactions between solitary waves can lead to complex behaviors.

Differences between Serre and Euler solutions are characterized.

Abstract

We numerically study nonlinear phenomena related to the dynamics of traveling wave solutions of the Serre equations including the stability, the persistence, the interactions and the breaking of solitary waves. The numerical method utilizes a high-order finite-element method with smooth, periodic splines in space and explicit Runge-Kutta methods in time. Other forms of solutions such as cnoidal waves and dispersive shock waves are also considered. The differences between solutions of the Serre equations and the Euler equations are also studied.