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 uxxxx​ and uxxtt​. We
obtain an explicit condition in terms of a, b and p 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=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 both
analytical and numerical results on the variation of the stability region of
wave velocities with a, b and p and then state explicitly the conditions
under which the traveling waves are orbitally stable.Comment: 16 pages, 4 figure