Instability and stability properties of traveling waves for the double dispersion equation. (March 2016) - Erbay, H.A.; Erbay, S.; Erkip, A.
Abstract: In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation u t t − u x x + a u x x x x − b u x x t t = − ( | u | p − 1 u ) x x 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 x x x x and u x x t t . 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 analytical as well as 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.