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

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.