Numerical Analysis and Approximate Travelling Wave Solutions for a Higher Order Internal Wave System

Abstract

In this work we focus on the numerical solution of a higher order bidirectional nonlinear model of Boussinesq type involving a nonlocal operator. Based on a von Neumann stability analysis for the linearized problem, an efficient and stable scheme for the nonlinear system is proposed. Our method is based on a numerical scheme known from the literature that solves satisfactorily a lower order linear system. Additionally, approximate periodic travelling wave solutions profiles for the higher order nonlinear system are presented. Such approximate travelling wave solutions are obtained from a solitary wave family of solutions for the Intermediate Long Wave (ILW) equation and the regularized Intermediate Long Wave (rILW) equation.

Travelling wave solutions for an internal wave model

“…The nonlinearity parameter is also nondimensional and assuming a weakly nonlinear wave propagation regime with small absolute wave amplitude if compared with h 1 , in such a way that α is of the same order of β, (α = O(β)), additional asymptotic expansions of the variables involved in system (1.1) can be performed. That is, by scaling η = αη * , u = αu * , and omitting the asterisks, the following dispersive, weakly nonlinear reduced model was derived from the strongly nonlinear system (1.1) in [13] and also in [25,26], after discarding terms of order α √ β, αβ and so on:…”