Dynamic transitions of generalized Kuramoto-Sivashinsky equation

Kiah Wah  Ong

doi:10.3934/dcdsb.2016.21.1225

Article Contents

2016, Volume 21, Issue 4: 1225-1236. Doi: 10.3934/dcdsb.2016.21.1225

This issue Previous Article A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations Next Article Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity

Dynamic transitions of generalized Kuramoto-Sivashinsky equation

Kiah Wah Ong^1,

1.
Department of Mathematics, Indiana University, Bloomington, IN 47405

Received: December 31, 2014

Revised: August 31, 2015

Published: May 2016

Abstract Related Papers Cited by

Abstract

In this article, we study the dynamic transition for the one dimensional generalized Kuramoto-Sivashinsky equation with periodic condition. It is shown that if the value of the dispersive parameter $\nu$ is strictly greater than $\nu^{\ast}$, then the transition is Type-I (continuous) and the bifurcated periodic orbit is an attractor as the control parameter $\lambda$ crosses the critical value $\lambda_0$. In the case where $\nu$ is strictly less than $\nu^{\ast}$, then the transition is Type-II (jump) and the trivial solution bifurcates to a unique unstable periodic orbit as the control parameter $\lambda$ crosses the critical value $\lambda_0$. The value of $\nu^{\ast}$ is also calculated in this paper.

Keywords:

Generalized Kuramoto-Sivashinsky equation,
centre manifold reduction,
phase transition dynamics,
spatial-temporal patterns.

Mathematics Subject Classification: Primary: 35Q35; Secondary: 35Q53.

Citation:

Related Papers

Cited by

References

[1]	B. Barker, M. A. Johnson, P. Noble, L. M. Rodrigues and K. Zumbrun, Nonlinear modulational stability of periodic traveling-wave solutions of the generalized kuramoto-sivashinsky equation, Physica D, 258 (2013), 11-46.doi: 10.1016/j.physd.2013.04.011.
[2]	H. Dijkstra, T. Sengul and S. Wang, Dynamic transitions of surface tension driven convection, Physica D, 247 (2013), 7-17.doi: 10.1016/j.physd.2012.12.008.
[3]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, New York, 1981.
[4]	A. P. Hooper and R. Grimshaw, Nonlinear instabilitity at the interface between two viscous fluids, Phys. Fluids, 28 (1985), 37-45.doi: 10.1063/1.865160.
[5]	Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theo. Phys., 55 (1976), 356-369.doi: 10.1143/PTP.55.356.
[6]	T. Ma and S. Wang, Stability and Bifurcation of Nonlinear Evolutions Equations, Science Press, Beijing, 2007.
[7]	T. Ma and S. Wang, Cahn-hilliard equations and phase transition dynamics for binary system, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 741-784.doi: 10.3934/dcdsb.2009.11.741.
[8]	T. Ma and S. Wang, Phase separation of binary systems, Physica A, 388 (2009), 4811-4817.doi: 10.1016/j.physa.2009.07.044.
[9]	T. Ma and S. Wang, Dynamic model and phase transitions for liquid helium, J. Math. Phys., 49 (2008), 073304, 18 pp.doi: 10.1063/1.2957943.
[10]	T. Ma and S. Wang, Dynamic bifurcation and stability in the rayleigh-benard convection, Commun. Math. Sci., 2 (2004), 159-183.doi: 10.4310/CMS.2004.v2.n2.a2.
[11]	T. Ma and S. Wang, Phase transitions for belousov-zhabotinsky reactions, Math. Methods Appl. Sci., 34 (2011), 1381-1397.doi: 10.1002/mma.1446.
[12]	T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.doi: 10.1142/9789812701152.
[13]	T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014.doi: 10.1007/978-1-4614-8963-4.
[14]	S. Wang and P. Yang, Remarks on the rayleigh-benard convection on spherical shells, J. Math. Fluid Mech., 15 (2013), 537-552.doi: 10.1007/s00021-012-0128-8.
[15]	G. I. Sivashinsky, On flame propagation under conditions of stoichiometry, SIAM J. Appl. Math, 39 (1980), 67-82.doi: 10.1137/0139007.
[16]	G. I. Sivashinsky, Instabilities, pattern-formation and turbulence in flames, Annu. Rev. Fluid Mech., 15 (1983), 179-199.doi: 10.1146/annurev.fl.15.010183.001143.