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

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• 作者：H.A. Erbay^a ; ^{husnuata.erbay@ozyegin.edu.tr" class="auth_mail" title="E-mail the corresponding author} ; S. Erbay^a ; ^{saadet.erbay@ozyegin.edu.tr" class="auth_mail" title="E-mail the corresponding author} ; A. Erkip^b ; ^{albert@sabanciuniv.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：74J35 ; 76B25 ; 35L70 ; 35Q51
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：133
• 期：Complete
• 页码：1-14
• 全文大小：936 K

文摘

In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation $u_{tt}-u_{xx}+a{u}_{xxxx}-b{u}_{xxtt}=-{\left({\left|u\right|}^{p-1}u\right)}_{xx}$ 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$u_{xxxx}$ and u_xxtt$u_{xxtt}$. We obtain an explicit condition in terms of a$a$, b$b$ and p$p$ 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$b=0$), 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$a$, b$b$ and p$p$ and then state explicitly the conditions under which the traveling waves are orbitally stable.