American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2021003

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

Kevin Li

260 Dwight St, New Haven, CT 06511, USA

Received  August 2020 Revised  September 2020 Published  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.

Keywords: Dynamic transitions, center manifold, bifurcation, Swift-Hohenberg equation, continuous and catastrophic transition.
Mathematics Subject Classification: Primary: 37G15, 35B36.
Citation: Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

T. Ma and S. Wang, Bifurcation and stability of superconductivity, J. Math. Phys., 46 (2005), 095112, 31 pp. doi: 10.1063/1.2012128.  Google Scholar

[6]

T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, Singapore, 2005. doi: 10.1142/5798.  Google Scholar

[7]

T. Ma and S. Wang, Phase Transition Dynamics, Springer Nature Switzerland AG, 2013. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

T. Ma and S. Wang, Bifurcation and stability of superconductivity, J. Math. Phys., 46 (2005), 095112, 31 pp. doi: 10.1063/1.2012128.  Google Scholar

[6]

T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, Singapore, 2005. doi: 10.1142/5798.  Google Scholar

[7]

T. Ma and S. Wang, Phase Transition Dynamics, Springer Nature Switzerland AG, 2013. Google Scholar

[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.  Google Scholar

Figure 1.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  A visual of the partition. $ \mathcal I_1 $, $ \mathcal I_2 $, $ \mathcal I_3 $, and $ \mathcal I_4 $ are encoded by the different shades
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Phase diagram for $ k = 2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Phase diagram for $ k = 6 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The phase diagram at $ \ell = 2\pi $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Forward in time trajectories (left) tending towards the stable periodic orbit, and backward in time trajectories (right) tending towards the unstable periodic orbit
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jongmin Han, Masoud Yari. Dynamic bifurcation of the complex Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 875-891. doi: 10.3934/dcdsb.2009.11.875
[2]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Doo Seok Lee. Bifurcation and final patterns of a modified Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2543-2567. doi: 10.3934/dcdsb.2017087
[3]

Toshiyuki Ogawa, Takashi Okuda. Bifurcation analysis to Swift-Hohenberg equation with Steklov type boundary conditions. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 273-297. doi: 10.3934/dcds.2009.25.273
[4]

J. Burke, Edgar Knobloch. Multipulse states in the Swift-Hohenberg equation. Conference Publications, 2009, 2009 (Special) : 109-117. doi: 10.3934/proc.2009.2009.109
[5]

Peng Gao. Averaging principles for the Swift-Hohenberg equation. Communications on Pure & Applied Analysis, 2020, 19 (1) : 293-310. doi: 10.3934/cpaa.2020016
[6]

Jongmin Han, Chun-Hsiung Hsia. Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2431-2449. doi: 10.3934/dcdsb.2012.17.2431
[7]

Ling-Jun Wang. The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1129-1156. doi: 10.3934/cpaa.2012.11.1129
[8]

Yanfeng Guo, Jinqiao Duan, Donglong Li. Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1701-1715. doi: 10.3934/dcdss.2016071
[9]

Yixia Shi, Maoan Han. Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3189-3204. doi: 10.3934/dcdss.2020114
[10]

Shengfu Deng. Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1647-1662. doi: 10.3934/dcdss.2016068
[11]

John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170
[12]

Masoud Yari. Attractor bifurcation and final patterns of the n-dimensional and generalized Swift-Hohenberg equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 441-456. doi: 10.3934/dcdsb.2007.7.441
[13]

Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure & Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219
[14]

Kiah Wah Ong. Dynamic transitions of generalized Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1225-1236. doi: 10.3934/dcdsb.2016.21.1225
[15]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
[16]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839
[17]

Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213
[18]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873
[19]

Stefano Bianchini, Alberto Bressan. A center manifold technique for tracing viscous waves. Communications on Pure & Applied Analysis, 2002, 1 (2) : 161-190. doi: 10.3934/cpaa.2002.1.161
[20]

Kelum Gajamannage, Erik M. Bollt. Detecting phase transitions in collective behavior using manifold's curvature. Mathematical Biosciences & Engineering, 2017, 14 (2) : 437-453. doi: 10.3934/mbe.2017027

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (30)
  • HTML views (140)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences