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Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion
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The Swift-Hohenberg equation is ubiquitous in the study of bistable dynamics. In this paper, we study the dynamic transitions of the Swift-Hohenberg equation with a third-order dispersion term in one spacial dimension with a periodic boundary condition. As a control parameter crosses a critical value, the trivial stable equilibrium solution will lose its stability, and undergoes a dynamic transition to a new physical state, described by a local attractor. The main result of this paper is to fully characterize the type and detailed structure of the transition using dynamic transition theory [
References:
[1] |
J. Han and C.-H. Hsia,
Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition, Discrete Contin. Dyn. Syst. Ser. B, 7 (2012), 2431-2449.
doi: 10.3934/dcdsb.2012.17.2431. |
[2] |
A. Hariz, L. Bahloul, L. Cherbi, K. Panajotov, M. Clerc, M. A. Ferré, B. Kostet, E. Averlant and M. Tlidi, Swift-Hohenberg equation with third-order dispersion for optical fiber resonators, Phys. Rev. A, 100 (2019), 023816.
doi: 10.1103/PhysRevA.100.023816. |
[3] |
T. Hoang and H. J. Hwang,
Dynamic pattern formation in Swift-Hohenberg equations, Quart. Appl. Math., 69 (2011), 603-612.
doi: 10.1090/S0033-569X-2011-01260-1. |
[4] |
C. Kieu, T. Sengul, Q. Wang and D. Yan,
On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents, Commun. Nonlinear Sci. Numer. Simul., 65 (2018), 196-215.
doi: 10.1016/j.cnsns.2018.05.010. |
[5] |
T. Ma and S. Wang, Bifurcation and stability of superconductivity, J. Math. Phys., 46 (2005), 095112, 31 pp.
doi: 10.1063/1.2012128. |
[6] |
T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, Singapore, 2005.
doi: 10.1142/5798. |
[7] |
T. Ma and S. Wang, Phase Transition Dynamics, Springer Nature Switzerland AG, 2013. |
[8] |
T. Şengül and S. Wang,
Dynamic transitions and baroclinic instability for 3D continuously stratified Boussinesq flows, J. Math. Fluid Mech., 20 (2018), 1173-1193.
doi: 10.1007/s00021-018-0361-x. |
show all references
References:
[1] |
J. Han and C.-H. Hsia,
Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition, Discrete Contin. Dyn. Syst. Ser. B, 7 (2012), 2431-2449.
doi: 10.3934/dcdsb.2012.17.2431. |
[2] |
A. Hariz, L. Bahloul, L. Cherbi, K. Panajotov, M. Clerc, M. A. Ferré, B. Kostet, E. Averlant and M. Tlidi, Swift-Hohenberg equation with third-order dispersion for optical fiber resonators, Phys. Rev. A, 100 (2019), 023816.
doi: 10.1103/PhysRevA.100.023816. |
[3] |
T. Hoang and H. J. Hwang,
Dynamic pattern formation in Swift-Hohenberg equations, Quart. Appl. Math., 69 (2011), 603-612.
doi: 10.1090/S0033-569X-2011-01260-1. |
[4] |
C. Kieu, T. Sengul, Q. Wang and D. Yan,
On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents, Commun. Nonlinear Sci. Numer. Simul., 65 (2018), 196-215.
doi: 10.1016/j.cnsns.2018.05.010. |
[5] |
T. Ma and S. Wang, Bifurcation and stability of superconductivity, J. Math. Phys., 46 (2005), 095112, 31 pp.
doi: 10.1063/1.2012128. |
[6] |
T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, Singapore, 2005.
doi: 10.1142/5798. |
[7] |
T. Ma and S. Wang, Phase Transition Dynamics, Springer Nature Switzerland AG, 2013. |
[8] |
T. Şengül and S. Wang,
Dynamic transitions and baroclinic instability for 3D continuously stratified Boussinesq flows, J. Math. Fluid Mech., 20 (2018), 1173-1193.
doi: 10.1007/s00021-018-0361-x. |








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