Transversal spectral instability of periodic traveling waves for the generalized Zakharov-Kuznetsov equation

TL;DR

This paper analyzes the spectral transversal stability of periodic traveling wave solutions in the two-dimensional generalized Zakharov-Kuznetsov equation, showing that all positive and certain sign-changing waves are spectrally unstable.

Contribution

It adapts existing spectral instability arguments to the periodic setting and characterizes the instability of all positive and some sign-changing periodic waves.

Findings

All positive periodic waves are spectrally unstable.

Sign-changing waves are also unstable under certain spectral conditions.

The method extends previous instability results to the periodic context.

Abstract

In this paper, we determine the transversal instability of periodic traveling wave solutions of the generalized Zakharov-Kuznetsov equation in two space dimensions. Using an adaptation of the arguments in \cite{nikolay} in the periodic context, it is possible to prove that all positive and one-dimensional $L-$periodic waves are spectrally (transversally) unstable. In addition, when periodic sign-changing waves exist, we also obtain the same property when the associated projection operator defined in the zero mean Sobolev space has only one negative eigenvalue.