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December  2021, 26(12): 6069-6090. doi: 10.3934/dcdsb.2021003

## Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion

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Received  August 2020 Revised  September 2020 Published  December 2021 Early access  December 2020

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 [7]. In particular, employing techniques from center manifold theory, we reduce this infinite dimensional problem to a finite one since the space on which the exchange of stability occurs is finite dimensional. The problem then reduces to analysis of single or double Hopf bifurcations, and we completely classify the possible phase changes depending on the dispersion for every spacial period.

Citation: Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6069-6090. doi: 10.3934/dcdsb.2021003
##### 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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##### 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.
These vector fields show the general shape of the case ${\mathop{\rm Re}\nolimits} C > 0$ and ${\mathop{\rm Re}\nolimits} D < 0$. The vertical axis represents $\rho_2$ and the horizontal represents $\rho_1$. The darker line is given by $\rho_2 = m_1 \rho_1$ and the lighter line is given by $\rho_2 = m_2 \rho_1$
Plot of ${\mathop{\rm Re}\nolimits} \beta_n(0)$ for various $\ell$. One can see from the figure that for large $\ell$, the maximas over $n \in \mathbb{Z}$ are attained at one or two pairs of conjugate eigenvalues. For sufficiently small $\ell$, $\beta_0(0)$ will be the maximum eigenvalue
A visual of the partition. $\mathcal I_1$, $\mathcal I_2$, $\mathcal I_3$, and $\mathcal I_4$ are encoded by the different shades
Phase diagram for $k = 2$
Phase diagram for $k = 6$
The phase diagram at $\ell = 2\pi$
Forward in time trajectories (left) tending towards the stable periodic orbit, and backward in time trajectories (right) tending towards the unstable periodic orbit
The blue line is the numerical approximations of the radius of the limit cycles as a function of $\lambda$, and the lighter line is the analytical limiting behavior as $\lambda \to 0$
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