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

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 [F. Rousset et N. Tzvetkov, 2010] 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 waves that change their sign exist, we also obtain the same property when the associated projection operator defined in the zero mean Sobolev space has only one negative eigenvalue.