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  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. ChampneysP. 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. HakkaevM. 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

show all references

References:
[1]

A. R. ChampneysP. 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. HakkaevM. 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

Figure 1.  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.
Figure 2.  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 $w = 0.8$.
Figure 3.  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 $\omega\in(0.82, 1)$.
Figure 4.  (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.
Figure 5.  Space-time evolution of the standing wave for $L = 30\pi$ (a) $p = 3$, $\omega = -0.55$ (b) $p = 5$, $\omega = -0.65$
Figure 6.  (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.
Figure 7.  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.
Figure 8.  The first and the second minimum eigenvalues of $\mathcal{H}$ as L varies on $[5\pi, 31\pi]$ for $c = 0$, $c = 1$ and $c = 1.3$.
Figure 9.  $c^*$ versus $L$. In this figure, $L$ varies on $[5\pi, 200\pi]$. The numerical computations show us as $L$ increases $c^*$ decreases.
Figure 10.  (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.
Figure 11.  (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.
Table 1.  $\omega^*$ 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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