Global bifurcation of solitary waves to the Boussinesq $abcd$ system

TL;DR

This paper investigates the existence and qualitative properties of solitary wave solutions to the Boussinesq $abcd$ system in non-Hamiltonian regimes, using analytic bifurcation techniques to understand global solution curves.

Contribution

It proves the existence of solitary waves in parameter regimes without Hamiltonian structure and analyzes their global bifurcation behavior and qualitative properties.

Findings

Existence of solitary waves bifurcating from stationary waves.

Global solution curves exhibit loss of ellipticity.

Solutions near classical supercritical waves may develop stagnation points.

Abstract

The Boussinesq $abcd$ system arises in the modeling of long wave small amplitude water waves in a channel, where the four parameters $(a,b,c,d)$ satisfy one constraint. In this paper we focus on the solitary wave solutions to such a system. In particular we work in two parameter regimes where the system does not admit a Hamiltonian structure (corresponding to $b \ne d$). We prove via analytic global bifurcation techniques the existence of solitary waves in such parameter regimes. Some qualitative properties of the solutions are also derived, from which sharp results can be obtained for the global solution curves. Specifically, we first construct solutions bifurcating from the stationary waves, and obtain a global continuous curve of solutions that exhibits a loss of ellipticity in the limit. The second family of solutions bifurcate from the classical Boussinesq supercritical waves. We show that the curve associated to the second class either undergoes a loss of ellipticity in the limit or becomes arbitrarily close to having a stagnation point.