Spectral stability of periodic waves in the generalized reduced Ostrovsky equation

Anna Geyer,Dmitry E Pelinovsky

doi:10.1007/s11005-017-0941-3

Abstract

We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations. Compared to the recent literature, we give a simple argument that proves spectral stability of all smooth periodic travelling waves independent of the nonlinearity power. The argument is based on the energy convexity and does not use coordinate transformations of the reduced Ostrovsky equations to the semi-linear equations of the Klein–Gordon type.

Highlights

We address the generalized reduced Ostrovsky equation written in the formx = u, (1)
Pelinovsky where p ∈ N is the nonlinearity power and u is a real-valued function of (x, t). This equation was derived in the context of long surface and internal gravity waves in a rotating fluid for p = 1 [22] and p = 2 [7]
These two cases are the only cases, for which the reduced Ostrovsky equation is transformed to integrable semi-linear equations of the Klein–Gordon type by means of a change of coordinates [6,14]

Summary

Introduction

The existence and spectral stability of smooth periodic travelling waves of the generalized reduced Ostrovsky equation (1) is summarized in the following theorem. As a limitation of the results of Theorem 1, we mention that the nonlinear orbital stability of travelling periodic waves cannot be established for the reduced Ostrovsky equations (1) by using the energy function (5) in space (6). This is because the local solution is defined in. Since T (E) < 0 for every E > 0, we have that T1 < π

Continuation of smooth periodic waves with respect to c

L 2per

Negative index of the operator L