# American Institute of Mathematical Sciences

• Previous Article
Dynamics of heterogeneous populations and communities and evolution of distributions
• PROC Home
• This Issue
• Next Article
Mean square approximation of multi dimensional reflecting fractional Brownian motion via penalty method
2005, 2005(Special): 476-486. doi: 10.3934/proc.2005.2005.476

## Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations

 1 Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece 2 Department of Telecommunications Science and Technology, University of the Peloponesse, Tripolis 22100, Greece 3 Department of Statistics and Actuarial Science, University of the Aegean, Karlovassi 83200, Samos, Greece

Received  September 2004 Revised  March 2005 Published  September 2005

We study the asymptotic behavior of complex discrete evolution equations of Ginzburg- Landau type. Depending on the nonlinearity and the data of the problem, we find different dynamical behavior ranging from global existence of solutions and global attractors, to blow up in finite time. We provide estimates for the blow up time, depending not only on the initial data but also on the size of the lattice. The theoretical estimates, are tested by numerical simulations.
Citation: N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934/proc.2005.2005.476
 [1] Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825 [2] Alessia Berti, Valeria Berti, Ivana Bochicchio. Global and exponential attractors for a Ginzburg-Landau model of superfluidity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 247-271. doi: 10.3934/dcdss.2011.4.247 [3] Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2021-2038. doi: 10.3934/cpaa.2021056 [4] N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711 [5] Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934/dcdsb.2019046 [6] Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229 [7] Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801 [8] Lu Zhang, Aihong Zou, Tao Yan, Ji Shu. Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021063 [9] Gregory A. Chechkin, Vladimir V. Chepyzhov, Leonid S. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1133-1154. doi: 10.3934/dcdsb.2018145 [10] Michael Stich, Carsten Beta. Standing waves in a complex Ginzburg-Landau equation with time-delay feedback. Conference Publications, 2011, 2011 (Special) : 1329-1334. doi: 10.3934/proc.2011.2011.1329 [11] Shijin Ding, Qiang Du. The global minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films. Communications on Pure & Applied Analysis, 2002, 1 (3) : 327-340. doi: 10.3934/cpaa.2002.1.327 [12] Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713 [13] N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647 [14] Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871 [15] Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173 [16] Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665 [17] Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359 [18] Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311 [19] Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 803-813. doi: 10.3934/cpaa.2009.8.803 [20] Xiaoying Han. Exponential attractors for lattice dynamical systems in weighted spaces. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 445-467. doi: 10.3934/dcds.2011.31.445

Impact Factor: