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

Erbay, H. A. and Erbay, S. and Erkip, Albert (2016) Instability and stability properties of traveling waves for the double dispersion equation. Nonlinear Analysis: Theory, Methods & Applications, 133 . pp. 1-14. ISSN 0362-546X (Print) 1873-5215 (Online)

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

Abstract

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.

Item Type:Article
Uncontrolled Keywords:Double dispersion equation; Instability by blow-up; Orbital stability
Subjects:Q Science > QA Mathematics > QA299.6-433 Analysis
ID Code:29279
Deposited By:Albert Erkip
Deposited On:20 May 2016 15:37
Last Modified:20 May 2016 15:37

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