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Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces
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.
References:
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Solitary waves in nonlinear beam equations: stability, fission and fusion, Nonlinear Dynam, 21 (2000), 31-53.
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The existence of ground states for fourth-order wave equations, Nonlinear Anal, 73 (2010), 367-373.
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Stability and instability of fourth order solitary waves, J. Dynamics and Differential Equations, 10 (1998), 151-188.
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P. J. McKenna and W. Walter,
Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715.
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Linear stability analysis for traveling waves of second order in time PDE's, Nonlinearity, 25 (2012), 2625-2654.
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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. |











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