June  2016, 21(4): 1225-1236. doi: 10.3934/dcdsb.2016.21.1225

Dynamic transitions of generalized Kuramoto-Sivashinsky equation

1. 

Department of Mathematics, Indiana University, Bloomington, IN 47405, United States

Received  January 2015 Revised  September 2015 Published  March 2016

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.
Citation: Kiah Wah Ong. Dynamic transitions of generalized Kuramoto-Sivashinsky equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1225-1236. doi: 10.3934/dcdsb.2016.21.1225
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.

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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.

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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.

show all references

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.

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