The Regularized Boussinesq equation: Instability of periodic traveling waves

Abstract

In this work we study the linear instability of periodic traveling waves associated with a generalization of the Regularized Boussinesq equation. 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 is obtained. With respect to applications of this approach, we prove the linear/nonlinear instability of cnoidal wave solutions for the modified Regularized Boussinesq equation and for a system of two coupled Boussinesq equations.