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

Aslihan Demirkaya; Milena Stanislavova

doi:10.3934/dcdsb.2018097

Article Contents

2019, Volume 24, Issue 1: 197-209. Doi: 10.3934/dcdsb.2018097

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

Aslihan Demirkaya^1, and
Milena Stanislavova^2,

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: December 31, 2016

Revised: June 30, 2017

Early access: March 2018

Published: January 2019

Stanislavova supported in part by NSF-DMS # 1516245.

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Abstract

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.

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Mathematics Subject Classification: Primary: 34K28, 34L16, 34D20, 34L16, 35B35, 35C07.

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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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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 $w = 0.8$.

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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 $\omega\in(0.82, 1)$.

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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.

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Figure 5. Space-time evolution of the standing wave for $L = 30\pi$ (a) $p = 3$, $\omega = -0.55$ (b) $p = 5$, $\omega = -0.65$

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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.

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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.

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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 $c = 1.3$.

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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.

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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.

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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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Table 1. $\omega^*$ values as $L$ varies.

$p$	$\omega^*$	$L$
$3$	$0.715\pm0.005$	$\pi$
	$0.655\pm0.005$	$\in[2\pi, 50\pi]$
	$0.6375\pm0.0025$	$100\pi$
$5$	$0.865\pm0.005$	$\pi$
	$0.825\pm0.005$	$\in[2\pi, 50\pi]$
	$0.8175\pm0.0025$	$100\pi$

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