Existence, regularity, and symmetry of periodic traveling waves for Gardner-Ostrovsky type equations

TL;DR

This paper investigates the existence, regularity, and symmetry of periodic traveling wave solutions for Gardner-Ostrovsky type equations, revealing how dispersion influences smoothness and employing novel symmetry proof techniques.

Contribution

It establishes the existence and regularity of solutions using bifurcation theory and introduces new symmetry results without traditional monotonicity assumptions.

Findings

Bifurcation theory proves existence of solutions with wave speed as parameter

Presence of Boussinesq dispersion ensures smooth solutions

Absence of dispersion can lead to peaked or cusped solutions

Abstract

We study the existence, regularity, and symmetry of periodic traveling solutions to a class of Gardner-Ostrovsky type equations, including the classical Gardner-Ostrovsky equation, the (modified) Ostrovsky, and the reduced (modified) Ostrovsky equation. The modified Ostrovsky equation is also known as the short pulse equation. The Gardner-Ostrovsky equation is a model for internal ocean waves of large amplitude. We prove the existence of nontrivial, periodic traveling wave solutions using local bifurcation theory, where the wave speed serves as the bifurcation parameter. Moreover, we give a regularity analysis for periodic traveling solutions in the presence as well as absence of Boussinesq dispersion. We see that the presence of Boussinesq dispersion implies smoothness of periodic traveling wave solutions, while its absence may lead to singularities in the form of peaks or cusps. Eventually, we study the symmetry of periodic traveling solutions by the method of moving planes. A novel feature of the symmetry results in the absence of Boussinesq dispersion is that we do not need to impose a traditional monotonicity condition or a recently developed reflection criterion on the wave profiles to prove the statement on the symmetry of periodic traveling waves.