On solitary-wave solutions of Boussinesq/Boussinesq systems for internal waves

Abstract

In this paper we consider a three-parameter system of Boussinesq/Boussinesq type, modeling the propagation of internal waves. Some theoretical and numerical properties of the systems were previously analyzed by the authors. As a second part of the study, the present paper is concerned with the analysis 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 based on a Fourier collocation approximation for the ode system of the solitary wave profiles with periodic boundary conditions, and on the iterative solution of the resulting fixed-point equations with the Petviashvili scheme combined with vector extrapolation techniques, is used to generate numerically approximations of solitary waves. These are an essential part of a computational study of the dynamics of the solitary waves, both classical and generalized. Using a full discretization based on spectral approximation in space of the corresponding periodic initial-value problem for the systems, and a fourth-order Runge–Kutta method of composition type as time integrator, we explore the evolution of small and large perturbations of the computed solitary-wave profiles, and we study computationally the collisions of solitary waves as well as the resolution of initial data into trains of solitary waves.