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Journal articles on the topic 'Complex Ginzburg-Landau equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Gao, Hongjun, and Keng-Huat Kwek. "Global existence for the generalised 2D Ginzburg-Landau equation." ANZIAM Journal 44, no. 3 (January 2003): 381–92. http://dx.doi.org/10.1017/s1446181100008099.

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AbstractGinzburg-Landau type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. Most work so far has concentrated on Ginzburg-Landau type equations with one spatial variable (1D). In this paper, the authors study a complex generalised Ginzburg-Landau equation with two spatial variables (2D) and fifth-order and cubic terms containing derivatives. Based on detail analysis, sufficient conditions for the existence and uniqueness of global solutions are obtained.

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2

Huang, Chunyan. "On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/607028.

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We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation(1/2⩽α⩽1)in any spatial dimensionn⩾1with rough initial data. For1/2<α⩽1, we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spacesMp,11-2α(1⩽p⩽∞). Forα=1/2, we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data inB˙∞,10(ℝn)∩M∞,10(ℝn). The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroupe-a+it-Δαto overcome the derivative in the nonlinear term.

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3

Ipsen, M., F. Hynne, and P. G. Sørensen. "Amplitude Equations and Chemical Reaction–Diffusion Systems." International Journal of Bifurcation and Chaos 07, no. 07 (July 1997): 1539–54. http://dx.doi.org/10.1142/s0218127497001217.

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The paper discusses the use of amplitude equations to describe the spatio-temporal dynamics of a chemical reaction–diffusion system based on an Oregonator model of the Belousov–Zhabotinsky reaction. Sufficiently close to a supercritical Hopf bifurcation the reaction–diffusion equation can be approximated by a complex Ginzburg–Landau equation with parameters determined by the original equation at the point of operation considered. We illustrate the validity of this reduction by comparing numerical spiral wave solutions to the Oregonator reaction–diffusion equation with the corresponding solutions to the complex Ginzburg–Landau equation at finite distances from the bifurcation point. We also compare the solutions at a bifurcation point where the systems develop spatio-temporal chaos. We show that the complex Ginzburg–Landau equation represents the dynamical behavior of the reaction–diffusion equation remarkably well, sufficiently far from the bifurcation point for experimental applications to be feasible.

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4

García-Morales, Vladimir, and Katharina Krischer. "The complex Ginzburg–Landau equation: an introduction." Contemporary Physics 53, no. 2 (March 2012): 79–95. http://dx.doi.org/10.1080/00107514.2011.642554.

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5

Graham, R., and T. Tél. "Potential for the Complex Ginzburg-Landau Equation." Europhysics Letters (EPL) 13, no. 8 (December 15, 1990): 715–20. http://dx.doi.org/10.1209/0295-5075/13/8/008.

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6

Zubrzycki, Andrzej. "A complex Ginzburg-Landau equation in magnetism." Journal of Magnetism and Magnetic Materials 150, no. 2 (October 1995): L143—L145. http://dx.doi.org/10.1016/0304-8853(95)00391-6.

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7

Mirzazadeh, Mohammad, Mehmet Ekici, Abdullah Sonmezoglu, Mostafa Eslami, Qin Zhou, Abdul H. Kara, Daniela Milovic, Fayequa B. Majid, Anjan Biswas, and Milivoj Belić. "Optical solitons with complex Ginzburg–Landau equation." Nonlinear Dynamics 85, no. 3 (May 9, 2016): 1979–2016. http://dx.doi.org/10.1007/s11071-016-2810-5.

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8

Salete, Eduardo, Antonio M. Vargas, Ángel García, Mihaela Negreanu, Juan J. Benito, and Francisco Ureña. "Complex Ginzburg–Landau Equation with Generalized Finite Differences." Mathematics 8, no. 12 (December 20, 2020): 2248. http://dx.doi.org/10.3390/math8122248.

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In this paper we obtain a novel implementation for irregular clouds of nodes of the meshless method called Generalized Finite Difference Method for solving the complex Ginzburg–Landau equation. We derive the explicit formulae for the spatial derivative and an explicit scheme by splitting the equation into a system of two parabolic PDEs. We prove the conditional convergence of the numerical scheme towards the continuous solution under certain assumptions. We obtain a second order approximation as it is clear from the numerical results. Finally, we provide several examples of its application over irregular domains in order to test the accuracy of the explicit scheme, as well as comparison with other numerical methods.

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9

Rosier, Lionel, and Bing-Yu Zhang. "Null controllability of the complex Ginzburg–Landau equation." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 26, no. 2 (February 2009): 649–73. http://dx.doi.org/10.1016/j.anihpc.2008.03.003.

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10

Cazenave, Thierry, Flávio Dickstein, and Fred B. Weissler. "Standing waves of the complex Ginzburg–Landau equation." Nonlinear Analysis: Theory, Methods & Applications 103 (July 2014): 26–32. http://dx.doi.org/10.1016/j.na.2014.03.001.

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11

Aranson, Igor S., and Lorenz Kramer. "The world of the complex Ginzburg-Landau equation." Reviews of Modern Physics 74, no. 1 (February 4, 2002): 99–143. http://dx.doi.org/10.1103/revmodphys.74.99.

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12

Hendrey, Matthew, Keeyeol Nam, Parvez Guzdar, and Edward Ott. "Target waves in the complex Ginzburg-Landau equation." Physical Review E 62, no. 6 (December 1, 2000): 7627–31. http://dx.doi.org/10.1103/physreve.62.7627.

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13

Akhmediev, N., A. Ankiewicz, and J. Soto-Crespo. "Multisoliton Solutions of the Complex Ginzburg-Landau Equation." Physical Review Letters 79, no. 21 (November 1997): 4047–51. http://dx.doi.org/10.1103/physrevlett.79.4047.

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14

Matsuoka, Chihiro, and Kazuhiro Nozaki. "Vortex Dynamics of the Complex Ginzburg-Landau Equation." Journal of the Physical Society of Japan 61, no. 5 (May 15, 1992): 1429–32. http://dx.doi.org/10.1143/jpsj.61.1429.

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15

Goh, Ryan, and Arnd Scheel. "Triggered Fronts in the Complex Ginzburg Landau Equation." Journal of Nonlinear Science 24, no. 1 (October 4, 2013): 117–44. http://dx.doi.org/10.1007/s00332-013-9186-1.

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16

Xiao, Jinghua, Gang Hu, Junzhong Yang, and Jihua Gao. "Controlling Turbulence in the Complex Ginzburg-Landau Equation." Physical Review Letters 81, no. 25 (December 21, 1998): 5552–55. http://dx.doi.org/10.1103/physrevlett.81.5552.

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17

Okazawa, Noboru. "Operator theory in the complex Ginzburg-Landau equation." Sugaku Expositions 31, no. 2 (September 19, 2018): 143–67. http://dx.doi.org/10.1090/suga/432.

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18

Doelman, A. "Traveling waves in the complex Ginzburg-Landau equation." Journal of Nonlinear Science 3, no. 1 (December 1993): 225–66. http://dx.doi.org/10.1007/bf02429865.

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19

Bechouche, Philippe, and Ansgar Jüngel. "Inviscid Limits¶of the Complex Ginzburg–Landau Equation." Communications in Mathematical Physics 214, no. 1 (October 2000): 201–26. http://dx.doi.org/10.1007/s002200000263.

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20

Collet, Pierre, and Jean-Pierre Eckmann. "Extensive Properties of the Complex Ginzburg-Landau Equation." Communications in Mathematical Physics 200, no. 3 (February 1, 1999): 699–722. http://dx.doi.org/10.1007/s002200050546.

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21

Li, Fang, and Bo You. "Global attractors for the complex Ginzburg–Landau equation." Journal of Mathematical Analysis and Applications 415, no. 1 (July 2014): 14–24. http://dx.doi.org/10.1016/j.jmaa.2014.01.059.

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22

Battogtokh, D., and A. Mikhailov. "Controlling turbulence in the complex Ginzburg-Landau equation." Physica D: Nonlinear Phenomena 90, no. 1-2 (January 1996): 84–95. http://dx.doi.org/10.1016/0167-2789(95)00232-4.

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23

Luce, Benjamin P. "Homoclinic explosions in the complex Ginzburg-Landau equation." Physica D: Nonlinear Phenomena 84, no. 3-4 (July 1995): 553–81. http://dx.doi.org/10.1016/0167-2789(95)00047-8.

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24

Melbourne, Ian, and Guido Schneider. "Phase dynamics in the complex Ginzburg–Landau equation." Journal of Differential Equations 199, no. 1 (May 2004): 22–46. http://dx.doi.org/10.1016/j.jde.2003.11.004.

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25

Shimotsuma, Daisuke, Tomomi Yokota, and Kentarou Yoshii. "Cauchy problem for the complex Ginzburg-Landau type Equation with $L^{p}$-initial data." Mathematica Bohemica 139, no. 2 (2014): 353–61. http://dx.doi.org/10.21136/mb.2014.143860.

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26

Huang, Chun, and Zhao Li. "New Exact Solutions of the Fractional Complex Ginzburg–Landau Equation." Mathematical Problems in Engineering 2021 (February 9, 2021): 1–8. http://dx.doi.org/10.1155/2021/6640086.

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In this paper, we apply the complete discrimination system method to establish the exact solutions of the fractional complex Ginzburg–Landau equation in the sense of the conformable fractional derivative. Firstly, by the fractional traveling wave transformation, time-space fractional complex Ginzburg–Landau equation is reduced to an ordinary differential equation. Secondly, some new exact solutions are obtained by the complete discrimination system method of the three-order polynomial; these solutions include solitary wave solutions, rational function solutions, triangle function solutions, and Jacobian elliptic function solutions. Finally, two numerical simulations are imitated to explain the propagation of optical pulses in optic fibers. At the same time, the comparison between the previous results and our results are also given.

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27

BLÖMKER, DIRK, and YONGQIAN HAN. "ASYMPTOTIC COMPACTNESS OF STOCHASTIC COMPLEX GINZBURG–LANDAU EQUATION ON AN UNBOUNDED DOMAIN." Stochastics and Dynamics 10, no. 04 (December 2010): 613–36. http://dx.doi.org/10.1142/s0219493710003121.

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The Ginzburg–Landau-type complex equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. In this paper, we consider the complex Ginzburg–Landau (CGL) equations on the whole real line perturbed by an additive spacetime white noise. Our main result shows that it generates an asymptotically compact stochastic or random dynamical system. This is a crucial property for the existence of a stochastic attractor for such CGL equations. We rely on suitable spaces with weights, due to the regularity properties of spacetime white noise, which gives rise to solutions that are unbounded in space.

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28

EGUÍLUZ, VÍCTOR M., EMILIO HERNÁNDEZ-GARCÍA, and ORESTE PIRO. "BOUNDARY EFFECTS IN THE COMPLEX GINZBURGàLANDAU EQUATION." International Journal of Bifurcation and Chaos 09, no. 11 (November 1999): 2209–14. http://dx.doi.org/10.1142/s0218127499001644.

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The effect of a finite geometry on the two-dimensional complex Ginzburg–Landau equation is addressed. Boundary effects induce the formation of novel states. For example, target-like solutions appear as robust solutions under Dirichlet boundary conditions. Synchronization of plane waves emitted by boundaries, entrainment by corner emission, and anchoring of defects by shock lines are also reported.

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29

G. Kaper, Hans, and Peter Takáč. "Bifurcating vortex solutions of the complex Ginzburg-Landau equation." Discrete & Continuous Dynamical Systems - A 5, no. 4 (1999): 871–80. http://dx.doi.org/10.3934/dcds.1999.5.871.

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30

Miot, Evelyne. "Dynamics of vortices for the complex Ginzburg–Landau equation." Analysis & PDE 2, no. 2 (May 1, 2009): 159–86. http://dx.doi.org/10.2140/apde.2009.2.159.

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31

Houghton, S. M., S. M. Tobias, E. Knobloch, and M. R. E. Proctor. "Bistability in the complex Ginzburg–Landau equation with drift." Physica D: Nonlinear Phenomena 238, no. 2 (January 2009): 184–96. http://dx.doi.org/10.1016/j.physd.2008.09.011.

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32

Xie, Ling-Ling, Jia-Zhen Gao, Wei-Miao Xie, and Ji-Hua Gao. "Amplitude wave in one-dimensional complex Ginzburg—Landau equation." Chinese Physics B 20, no. 11 (November 2011): 110503. http://dx.doi.org/10.1088/1674-1056/20/11/110503.

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33

Hong-Li, Wang, and Qi Ou-Yang. "Transition to Antispirals in the Complex Ginzburg–Landau Equation." Chinese Physics Letters 21, no. 8 (July 30, 2004): 1437–40. http://dx.doi.org/10.1088/0256-307x/21/8/007.

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34

Machihara, Shuji, and Yoshihisa Nakamura. "The inviscid limit for the complex Ginzburg–Landau equation." Journal of Mathematical Analysis and Applications 281, no. 2 (May 2003): 552–64. http://dx.doi.org/10.1016/s0022-247x(03)00143-4.

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35

Zubrzycki, A. "Turbulence in the One-Dimensional Complex Ginzburg-Landau Equation." Acta Physica Polonica A 87, no. 6 (June 1995): 925–32. http://dx.doi.org/10.12693/aphyspola.87.925.

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36

Hakim, Vincent, and Wouter-Jan Rappel. "Dynamics of the globally coupled complex Ginzburg-Landau equation." Physical Review A 46, no. 12 (December 1, 1992): R7347—R7350. http://dx.doi.org/10.1103/physreva.46.r7347.

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37

Doering, C. R., J. D. Gibbon, D. D. Holm, and B. Nicolaenko. "Low-dimensional behaviour in the complex Ginzburg-Landau equation." Nonlinearity 1, no. 2 (May 1, 1988): 279–309. http://dx.doi.org/10.1088/0951-7715/1/2/001.

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38

Wilson, R. Eddie. "Numerically derived scalings for the complex Ginzburg-Landau equation." Physica D: Nonlinear Phenomena 112, no. 3-4 (February 1998): 329–43. http://dx.doi.org/10.1016/s0167-2789(97)00181-4.

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39

Yomba, Emmanuel, and Timoléon C. Kofané. "On exact solutions of modified complex Ginzburg-Landau equation." Physica D: Nonlinear Phenomena 125, no. 1-2 (January 1999): 105–22. http://dx.doi.org/10.1016/s0167-2789(98)00152-3.

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40

Porubov, A. V., and M. G. Velarde. "Exact periodic solutions of the complex Ginzburg–Landau equation." Journal of Mathematical Physics 40, no. 2 (February 1999): 884–96. http://dx.doi.org/10.1063/1.532692.

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41

Cazenave, Thierry, Flávio Dickstein, and Fred B. Weissler. "Finite-Time Blowup for a Complex Ginzburg--Landau Equation." SIAM Journal on Mathematical Analysis 45, no. 1 (January 2013): 244–66. http://dx.doi.org/10.1137/120878690.

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42

Ilak, Miloš, Shervin Bagheri, Luca Brandt, Clarence W. Rowley, and Dan S. Henningson. "Model Reduction of the Nonlinear Complex Ginzburg–Landau Equation." SIAM Journal on Applied Dynamical Systems 9, no. 4 (January 2010): 1284–302. http://dx.doi.org/10.1137/100787350.

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43

Jun-Zhong, Yang, and Zhang Mei. "Drift of Spiral Waves in Complex Ginzburg–Landau Equation." Communications in Theoretical Physics 45, no. 4 (April 2006): 647–52. http://dx.doi.org/10.1088/0253-6102/45/4/016.

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44

Gao Ji-Hua, Xie Wei-Miao, Gao Jia-Zhen, Yang Hai-Peng, and Ge Zao-Chuan. "Amplitude spiral wave in coupled complex Ginzburg-Landau equation." Acta Physica Sinica 61, no. 13 (2012): 130506. http://dx.doi.org/10.7498/aps.61.130506.

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45

Faller, Roland, and Lorenz Kramer. "Phase chaos in the anisotropic complex Ginzburg-Landau equation." Physical Review E 57, no. 6 (June 1, 1998): R6249—R6252. http://dx.doi.org/10.1103/physreve.57.r6249.

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46

Wu, Jiahong. "The Inviscid Limit of the Complex Ginzburg–Landau Equation." Journal of Differential Equations 142, no. 2 (January 1998): 413–33. http://dx.doi.org/10.1006/jdeq.1997.3347.

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47

Yomba, Emmanuel, and Timoléon Crépin Kofané. "Solutions of the Lowest Order Complex Ginzburg-Landau Equation." Journal of the Physical Society of Japan 69, no. 4 (April 15, 2000): 1027–32. http://dx.doi.org/10.1143/jpsj.69.1027.

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48

Cheng, Ming. "Recurrent motion in the fractional complex Ginzburg–Landau equation." Journal of Mathematical Physics 61, no. 11 (November 1, 2020): 111507. http://dx.doi.org/10.1063/1.5131376.

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49

Aranson, Igor S., Hugues Chaté, and Lei-Han Tang. "Spiral Motion in a Noisy Complex Ginzburg-Landau Equation." Physical Review Letters 80, no. 12 (March 23, 1998): 2646–49. http://dx.doi.org/10.1103/physrevlett.80.2646.

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50

López, Vanessa, Philip Boyland, Michael T. Heath, and Robert D. Moser. "Relative Periodic Solutions of the Complex Ginzburg--Landau Equation." SIAM Journal on Applied Dynamical Systems 4, no. 4 (January 2005): 1042–75. http://dx.doi.org/10.1137/040618977.

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