Existence of multi-pulses of the regularized short-pulse and Ostrovsky equations

TL;DR

This paper proves the existence of multi-pulse solutions near orbit-flip bifurcations in singularly perturbed reversible systems, specifically applied to the regularized short-pulse and Ostrovsky equations, using geometric and analytical methods.

Contribution

It demonstrates the existence of multi-pulses in singularly perturbed systems near orbit-flip bifurcations, extending previous results to these specific equations.

Findings

Multi-pulses exist near orbit-flip bifurcations in the studied equations.

The sign of a geometric condition determines multi-pulse existence.

The methods combine geometric singular perturbation theory and Lyapunov--Schmidt reduction.

Abstract

The existence of multi-pulse solutions near orbit-flip bifurcations of a primary single-humped pulse is shown in reversible, conservative, singularly perturbed vector fields. Similar to the non-singular case, the sign of a geometric condition that involves the first integral decides whether multi-pulses exist or not. The proof utilizes a combination of geometric singular perturbation theory and Lyapunov--Schmidt reduction through Lin's method. The motivation for considering orbit flips in singularly perturbed systems comes from the regularized short-pulse equation and the Ostrovsky equation, which both fit into this framework and are shown here to support multi-pulses.