Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. II. Dynamics

TL;DR

This paper investigates the dynamics of solitary waves in internal wave models using numerical simulations, focusing on stability, collisions, wave resolution, and shock formation, extending previous work on solitary-wave profiles.

Contribution

It advances the understanding of internal wave solitary solutions by analyzing their dynamics through computational methods, including stability and interaction studies.

Findings

Solitary waves exhibit stability under small perturbations.

Wave collisions produce complex interaction patterns.

Initial data can resolve into trains of solitary waves.

Abstract

Considered here are two systems of equations modeling the two-way propagation of long-crested, long-wavelength internal waves along the interface of a two-layer system of fluids in the Benjamin-Ono and the Intermediate Long-Wave regime, respectively. These systems were previously shown to have solitary-wave solutions, decaying to zero algebraically for the Benjamin-Ono system, and exponentially in the Intermediate Long-Wave regime. Several methods to approximate solitary-wave profiles were introduced and analyzed by the authors in Part I of this project. A natural continuation of this previous work, pursued here, is to study the dynamics of the solitary-wave solutions of these systems. This will be done by computational means using a discretization of the periodic initial-value problem. The numerical method used here is a Fourier spectral method for the spatial approximation coupled with a fourth-order, explicit Runge-Kutta time stepping. The resulting, fully discrete scheme is used to study computationally the stability of the solitary waves under small and large perturbations, the collisions of solitary waves, the resolution of initial data into trains of solitary waves, and the formation of dispersive shock waves. Comparisons with related unidirectional models are also undertaken.