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.
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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 |
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.
Keywords:
Nonlinear beam equation,
standing waves,
traveling waves,
spectral stability,
orbital stability.
Mathematics Subject Classification:
Primary: 34K28, 34L16, 34D20, 34L16, 35B35, 35C07.
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
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References:
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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. |
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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. |
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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.
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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. |
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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. |
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S. Levandosky,
Stability and instability of fourth order solitary waves, J. Dynamics and Differential Equations, 10 (1998), 151-188.
doi: 10.1023/A:1022644629950. |
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P. J. McKenna and W. Walter,
Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715.
doi: 10.1137/0150041. |
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J. Smoller, Nonlinear Ordinary Differential Equations, CRC Press, Boca Raton, FL, 1993.
|
| [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. |
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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. |
show all references
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [8] |
J. Smoller, Nonlinear Ordinary Differential Equations, CRC Press, Boca Raton, FL, 1993.
|
| [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. |
| [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. |
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.
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