Exact internal waves of a Boussinesq system

TL;DR

This paper explicitly finds all traveling wave solutions of a Boussinesq system modeling internal waves at fluid boundaries, including several new solutions, using a method for solving algebraic ODEs.

Contribution

It provides a complete classification of elliptic and degenerate elliptic solutions for the Boussinesq system's traveling waves, including new solutions.

Findings

All solutions in the class are explicitly obtained.

Several new solutions are identified.

The method is applicable to similar nonlinear ODEs.

Abstract

We consider a Boussinesq system describing one-dimensional internal waves which develop at the boundary between two immiscible fluids, and we restrict to its traveling waves. The method which yields explicitly all the elliptic or degenerate elliptic solutions of a given nonlinear, any order algebraic ordinary differential equation is briefly recalled. We then apply it to the fluid system and, restricting in this preliminary report to the generic situation, we obtain all the solutions in that class, including several new solutions.