Instability and stability properties of traveling waves for the double dispersion equation

In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation utt−uxx+auxxxx−buxxtt=−(|u|p−1u)xx for p>1, a>b>0. The main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms uxxxxuxxxx and uxxttuxxtt. We obtain an explicit condition in terms of aa, bb and pp on wave velocities ensuring that traveling wave solutions of the double dispersion equation are strongly unstable by blow up. In the special case of the Boussinesq equation (b=0b=0), our condition reduces to the one given in the literature. For the double dispersion equation, we also investigate orbital stability of traveling waves by considering the convexity of a scalar function. We provide analytical as well as numerical results on the variation of the stability region of wave velocities with aa, bb and pp and then state explicitly the conditions under which the traveling waves are orbitally stable.