On the instability of periodic waves for dispersive equations

Abstract

This paper sheds new light on the linear instability of periodic traveling wave associated with some general one-dimensional dispersive models. By using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles with mean zero is obtained. Applications of this approach are concerning with the linear instability of cnoidal wave solutions for the modified Benjamin-Bona-Mahony and the modified Korteweg-de Vries equations. The arguments presented in this investigation has prospects for the study of the instability of periodic traveling wave of other nonlinear evolution equations.