Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation
Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation
复制标题
四阶梁方程驻波和行波的存在性和稳定性的数值结果
DOI：
10.3934/dcdsb.2018097
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
M. Stanislavova
中科院分区：
文献类型：
--
作者：
A. Demirkaya;M. Stanislavova
In this paper, we study numerically the existence and stability of some special solutions of the nonlinear beam equation: \begin{document}$u_{tt}+u_{xxxx}+u-|u|^{p-1} u = 0$\end{document} when \begin{document}$p = 3$\end{document} and \begin{document}$p = 5$\end{document} . For the standing wave solutions \begin{document}$u(x, t) = e^{iω t}\varphi_{ω}(x)$\end{document} we numerically illustrate their existence using variational approach. Our numerics illustrate the existence of both ground states and excited states. We also compute numerically the threshold value \begin{document}$ω^*$\end{document} which separates stable and unstable ground states. Next, we study the existence and linear stability of periodic traveling wave solutions \begin{document}$u(x, t) = φ_c(x+ct)$\end{document} . We present numerical illustration of the theoretically predicted threshold value of the speed \begin{document}$c$\end{document} which separates the stable and unstable waves.
In this paper, we study numerically the existence and stability of some special solutions of the nonlinear beam equation: \begin{document}$u_{tt}+u_{xxxx}+u-|u|^{p-1} u = 0$\end{document} when \begin{document}$p = 3$\end{document} and \begin{document}$p = 5$\end{document} . For the standing wave solutions \begin{document}$u(x, t) = e^{iω t}\varphi_{ω}(x)$\end{document} we numerically illustrate their existence using variational approach. Our numerics illustrate the existence of both ground states and excited states. We also compute numerically the threshold value \begin{document}$ω^*$\end{document} which separates stable and unstable ground states. Next, we study the existence and linear stability of periodic traveling wave solutions \begin{document}$u(x, t) = φ_c(x+ct)$\end{document} . We present numerical illustration of the theoretically predicted threshold value of the speed \begin{document}$c$\end{document} which separates the stable and unstable waves.