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

Erbay, H.A. and Erbay, S. and Erkip, Albert (2015) Instability and stability properties of traveling waves for the double dispersion equation. (Accepted/In Press)

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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
Subjects: Q Science > QA Mathematics > QA299.6-433 Analysis
Divisions: Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics
Faculty of Engineering and Natural Sciences
Depositing User: Albert Erkip
Date Deposited: 28 Nov 2015 15:14
Last Modified: 02 Aug 2019 12:18
URI: https://research.sabanciuniv.edu/id/eprint/27697

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