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

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Official URL: http://dx.doi.org/10.1016/j.na.2015.11.019

In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation u_tt −u_xx + au_xxxx − bu_xxtt = −(|u|^(p−1)u)_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 u_xxxx and u_xxtt. We obtain an explicit condition in terms of a, b and p on wave velocities ensuring that traveling wave solutionsof 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.