Notes on numerical analysis and solitary wave solutions of Boussinesq/Boussinesq systems for internal waves

TL;DR

This paper analyzes a three-parameter family of Boussinesq systems modeling internal waves, exploring their mathematical properties, numerical approximation, and the dynamics of solitary wave solutions through theoretical and computational methods.

Contribution

It provides new theoretical insights into well-posedness, conservation laws, and Hamiltonian structure, along with numerical methods for simulating solitary waves and their interactions.

Findings

Existence of classical and generalized solitary waves depending on parameters

Numerical generation and simulation of solitary wave dynamics

Collision and resolution behaviors of solitary waves studied computationally

Abstract

In this paper a three-parameter family of Boussinesq systems is studied. The systems have been proposed as models of the propagation of long internal waves along the interface of a two-layer system of fluids with rigid-lid condition for the upper layer and under a Boussinesq regime for the flow in both layers. We first present some theoretical properties of well-posedness, conservation laws and Hamiltonian structure of the systems. Then the corresponding periodic initial-value problem is discretized in space by the spectral Fourier Galerkin method and for each system, error estimates for the semidiscrete approximation are proved. The rest of the paper is concerned with the study of existence and the numerical simulation of some issues of the dynamics of solitary-wave solutions. Standard theories are used to derive several results of existence of classical and generalized solitary waves, depending on the parameters of the models. A numerical procedure is used to generate numerically approximations of solitary waves. These are taken as initial conditions in a computational study of the dynamics of the solitary waves, both classical and generalized. To this end, the spectral semidiscretizations of the periodic initial-value problem for the systems are numerically integrated by a fourth-order Runge-Kutta-composition method based on the implicit midpoint rule. The fully discrete scheme is then used to approximate the evolution of perturbations of computed solitary wave profiles, and to study computationally the collisions of solitary waves as well as the resolution of initial data into trains of solitary waves.