Spectral stability for classical periodic waves of the Ostrovsky and short pulse models

TL;DR

This paper demonstrates spectral stability of classical periodic waves in the Ostrovsky and short pulse models with periodic boundary conditions, using analysis of non-standard eigenvalue problems involving Hill operators.

Contribution

It provides a new spectral stability analysis for periodic waves in these models, including explicit construction of solutions for the short pulse model.

Findings

Spectral stability for all parameter values in both models.

Explicit construction of short pulse traveling waves using Jacobi elliptic functions.

Analysis of non-standard eigenvalue problems involving Hill operators.

Abstract

We consider the Ostrovsky and short pulse models in a symmetric spatial interval, subject to periodic boundary conditions. For the Ostrovsky case, we revisit the classical periodic traveling waves and for the short pulse model, we explicitly construct traveling waves in terms of Jacobi elliptic functions. For both examples, we show spectral stability, for all values of the parameters. This is achieved by studying the non-standard eigenvalue problems in the form $L u=\la u'$, where $L$ is a Hill operator.