Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation
| Autor: | A. Demirkaya, Milena Stanislavova |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Physics
Applied Mathematics 010102 general mathematics 01 natural sciences Stability (probability) 010101 applied mathematics Standing wave Fourth order Nonlinear beam Excited state Traveling wave Discrete Mathematics and Combinatorics 0101 mathematics Beam (structure) Linear stability Mathematical physics |
| Zdroj: | Discrete & Continuous Dynamical Systems - B. 24:197-209 |
| ISSN: | 1553-524X |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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