Instability and stability properties of traveling waves for the double dispersion equation. (March 2016)

• Record Type: Journal Article
• Authors: Erbay, H.A.; Erbay, S.; Erkip, A.
• Is Part Of: Nonlinear analysis. Volume 133(2016)
• Page Start: 1
• Page End: 14
• Publication Date: 2016-03
• Subjects: 74J35 -- 76B25 -- 35L70 -- 35Q51; Double dispersion equation -- Instability by blow-up -- Orbital stability; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.11.019
• Languages: English
• ISSNs: 0362-546X

Abstract

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.