# American Institute of Mathematical Sciences

January  2019, 24(1): 197-209. doi: 10.3934/dcdsb.2018097

## Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation

 1 Department of Mathematics, University of Hartford, 200 Bloomfield Avenue, West Hartford, CT 06117, USA 2 Department of Mathematics, University of Kansas, 1460 Jayhawk Boulevard, Lawrence KS 66045-7523, USA

Received  January 2017 Revised  July 2017 Published  January 2019 Early access  March 2018

Fund Project: Stanislavova supported in part by NSF-DMS # 1516245.

In this paper, we study numerically the existence and stability of some special solutions of the nonlinear beam equation: $u_{tt}+u_{xxxx}+u-|u|^{p-1} u = 0$ when $p = 3$ and $p = 5$. For the standing wave solutions $u(x, t) = e^{iω t}\varphi_{ω}(x)$ 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 $ω^*$ which separates stable and unstable ground states. Next, we study the existence and linear stability of periodic traveling wave solutions $u(x, t) = φ_c(x+ct)$. We present numerical illustration of the theoretically predicted threshold value of the speed $c$ which separates the stable and unstable waves.

Citation: Aslihan Demirkaya, Milena Stanislavova. Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 197-209. doi: 10.3934/dcdsb.2018097
##### References:
 [1] A. R. Champneys, P. J. McKenna and P. A. Zegeling, Solitary waves in nonlinear beam equations: stability, fission and fusion, Nonlinear Dynam, 21 (2000), 31-53.  doi: 10.1023/A:1008302207311.  Google Scholar [2] L. Chen, Orbital stability of solitary waves for the Klein-Gordon-Zakharov equations, Acta Math. Appl. Sinica, 15 (1999), 54-64.  doi: 10.1007/BF02677396.  Google Scholar [3] Y. Chen and P. J. McKenna, Traveling waves in a nonlinearly suspended beam: Theoretical results and numerical observations, J. Differential Equations, 136 (1997), 325-355.  doi: 10.1006/jdeq.1996.3155.  Google Scholar [4] S. Hakkaev, M. Stanislavova and A. Stefanov, Orbital Stability for periodic standing waves of the Klein-Gordon-Zakharov and the Beam equation, ZAMP-Zeitschrift fuer Angewandte Mathematik und Physik,, 64 (2013), 265-282.  doi: 10.1007/s00033-012-0228-6.  Google Scholar [5] P. Karageorgis and P. J. McKenna, The existence of ground states for fourth-order wave equations, Nonlinear Anal, 73 (2010), 367-373.  doi: 10.1016/j.na.2010.03.025.  Google Scholar [6] S. Levandosky, Stability and instability of fourth order solitary waves, J. Dynamics and Differential Equations, 10 (1998), 151-188.  doi: 10.1023/A:1022644629950.  Google Scholar [7] P. J. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715.  doi: 10.1137/0150041.  Google Scholar [8] J. Smoller, Nonlinear Ordinary Differential Equations, CRC Press, Boca Raton, FL, 1993.   Google Scholar [9] M. Stanislavova and A. Stefanov, Linear stability analysis for traveling waves of second order in time PDE's, Nonlinearity, 25 (2012), 2625-2654.  doi: 10.1088/0951-7715/25/9/2625.  Google Scholar [10] M. Stanislavova and A. Stefanov, Spectral stability analysis for special solutions of second order in time PDE's: the higher dimensional case, Physica D: Nonlinear Phenomena, 262 (2013), 1-13.  doi: 10.1016/j.physd.2013.06.014.  Google Scholar

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##### References:
 [1] A. R. Champneys, P. J. McKenna and P. A. Zegeling, Solitary waves in nonlinear beam equations: stability, fission and fusion, Nonlinear Dynam, 21 (2000), 31-53.  doi: 10.1023/A:1008302207311.  Google Scholar [2] L. Chen, Orbital stability of solitary waves for the Klein-Gordon-Zakharov equations, Acta Math. Appl. Sinica, 15 (1999), 54-64.  doi: 10.1007/BF02677396.  Google Scholar [3] Y. Chen and P. J. McKenna, Traveling waves in a nonlinearly suspended beam: Theoretical results and numerical observations, J. Differential Equations, 136 (1997), 325-355.  doi: 10.1006/jdeq.1996.3155.  Google Scholar [4] S. Hakkaev, M. Stanislavova and A. Stefanov, Orbital Stability for periodic standing waves of the Klein-Gordon-Zakharov and the Beam equation, ZAMP-Zeitschrift fuer Angewandte Mathematik und Physik,, 64 (2013), 265-282.  doi: 10.1007/s00033-012-0228-6.  Google Scholar [5] P. Karageorgis and P. J. McKenna, The existence of ground states for fourth-order wave equations, Nonlinear Anal, 73 (2010), 367-373.  doi: 10.1016/j.na.2010.03.025.  Google Scholar [6] S. Levandosky, Stability and instability of fourth order solitary waves, J. Dynamics and Differential Equations, 10 (1998), 151-188.  doi: 10.1023/A:1022644629950.  Google Scholar [7] P. J. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715.  doi: 10.1137/0150041.  Google Scholar [8] J. Smoller, Nonlinear Ordinary Differential Equations, CRC Press, Boca Raton, FL, 1993.   Google Scholar [9] M. Stanislavova and A. Stefanov, Linear stability analysis for traveling waves of second order in time PDE's, Nonlinearity, 25 (2012), 2625-2654.  doi: 10.1088/0951-7715/25/9/2625.  Google Scholar [10] M. Stanislavova and A. Stefanov, Spectral stability analysis for special solutions of second order in time PDE's: the higher dimensional case, Physica D: Nonlinear Phenomena, 262 (2013), 1-13.  doi: 10.1016/j.physd.2013.06.014.  Google Scholar
Two standing waves are shown for $p = 3$, $\omega = 0.5$ and $L = 20\pi$. The dashed line is the standing wave derived from a local minimizer of (8) and the solid line is derived from a global one.
Existence of standing waves. $\varphi_{\omega}$ versus position when $p = 3$, (a) for different values of $\omega$ for $L = 50\pi$ (b) for different values of $L$ for $w = 0.8$.
Orbital stability of standing wave solutions. $M(\omega)$ versus $\omega$ when $L = 50\pi$, (a) $p = 3$, the graph is concave up for $\omega\in (0.64, 1)$, (b) $p = 5$, the graph is concave up for $\omega\in(0.82, 1)$.
(a) Snap-shots from the simulation of a periodic standing wave for $p = 5$, $\omega = -0.95$, $L = 30\pi$ when $t = 0$ (blue), $t = 5$ (red), $t = 22$ (green), $t = 28$ (pink), $t = 39$ (purple), $t = 44$ (black). (b) the space-time evolution of the periodic standing wave.
Space-time evolution of the standing wave for $L = 30\pi$ (a) $p = 3$, $\omega = -0.55$ (b) $p = 5$, $\omega = -0.65$
(a) Snap-shots from the simulation of a periodic traveling wave for $c = -1, 32$, $L = 30\pi$ when $t = 0$ (blue), $t = 1$ (red) and $t = 50$ (green) (b) the space-time evolution of the periodic traveling wave.
Existence of traveling waves. $\phi_{c}$ versus position for different values of $c$ when $L = 100\pi$ and $p = 3$. $c = 0$ corresponds to the steady state solution.
The first and the second minimum eigenvalues of $\mathcal{H}$ as L varies on $[5\pi, 31\pi]$ for $c = 0$, $c = 1$ and $c = 1.3$.
$c^*$ versus $L$. In this figure, $L$ varies on $[5\pi, 200\pi]$. The numerical computations show us as $L$ increases $c^*$ decreases.
(a) Snap-shots from the simulation of a periodic traveling wave for $c = -1, 38$, $L = 30\pi$ when $t = 0$ (blue), $t = 1$ (red) and $t = 50$ (green) (b) the space-time evolution of the periodic traveling wave.
(a) Snap-shots from the simulation of a periodic standing wave for $p = 3$, $\omega = -0.85$, $L = 30\pi$ when $t = 0$ (blue), $t = 5$ (red), $t = 17$ (green), $t = 24$ (pink), $t = 37$ (purple), $t = 49$ (cyan), $t = 56$ (black). (b) the space-time evolution of the periodic standing wave.
$\omega^*$ values as $L$ varies.
 $p$ $\omega^*$ $L$ $3$ $0.715\pm0.005$ $\pi$ $0.655\pm0.005$ $\in[2\pi, 50\pi]$ $0.6375\pm0.0025$ $100\pi$ $5$ $0.865\pm0.005$ $\pi$ $0.825\pm0.005$ $\in[2\pi, 50\pi]$ $0.8175\pm0.0025$ $100\pi$
 $p$ $\omega^*$ $L$ $3$ $0.715\pm0.005$ $\pi$ $0.655\pm0.005$ $\in[2\pi, 50\pi]$ $0.6375\pm0.0025$ $100\pi$ $5$ $0.865\pm0.005$ $\pi$ $0.825\pm0.005$ $\in[2\pi, 50\pi]$ $0.8175\pm0.0025$ $100\pi$
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