American Institute of Mathematical Sciences

Advanced
November  2008, 7(6): 1345-1359. doi: 10.3934/cpaa.2008.7.1345

Numerical mountain pass solutions of Ginzburg-Landau type equations

Dmitry Glotov 1, and  P. J. McKenna 1,

1. 

Department of Mathematics, 196 Auditorium Road, Unit 3009, University of Connecticut, Storrs, CT 06269-3009, United States, United States

Received  August 2007 Revised  May 2008 Published  August 2008

We study the numerical solutions of a system of Ginzburg-Landau type equations arising in the thin film model of superconductivity. These solutions are obtained by the Mountain Pass algorithm that was originally developed for semilinear elliptic equations. We prove a key hypothesis of the Mountain Pass theorem and investigate the physical features of the solutions such as the presence, the number, and the location of vortices and the numerical properties such as stability.
Keywords: mountain pass method, Ginzburg-Landau equations, numerical solution..
Mathematics Subject Classification: Primary: 35J60, 35J50; Secondary: 65N9.
Citation: Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[1]

Kolade M. Owolabi, Edson Pindza. Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 835-851. doi: 10.3934/dcdss.2020048
[2]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205
[3]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713
[4]

Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934/proc.2001.2001.280
[5]

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
[6]

Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754
[7]

Bixiang Wang, Shouhong Wang. Gevrey class regularity for the solutions of the Ginzburg-Landau equations of superconductivity. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 507-522. doi: 10.3934/dcds.1998.4.507
[8]

Gregory A. Chechkin, Vladimir V. Chepyzhov, Leonid S. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1133-1154. doi: 10.3934/dcdsb.2018145
[9]

Lu Zhang, Aihong Zou, Tao Yan, Ji Shu. Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 749-768. doi: 10.3934/dcdsb.2021063
[10]

Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745
[11]

Linghua Kong, Liqun Kuang, Tingchun Wang. Efficient numerical schemes for two-dimensional Ginzburg-Landau equation in superconductivity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6325-6347. doi: 10.3934/dcdsb.2019141
[12]

Yueling Jia, Zhaohui Huo. Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation. Kinetic and Related Models, 2014, 7 (1) : 57-77. doi: 10.3934/krm.2014.7.57
[13]

Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks and Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715
[14]

Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121
[15]

Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934/dcdsb.2018095
[16]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
[17]

Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181
[18]

Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934/dcdsb.2019046
[19]

Hong Lu, Mingji Zhang. Dynamics of non-autonomous fractional Ginzburg-Landau equations driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3553-3576. doi: 10.3934/dcdsb.2020072
[20]

Dandan Ma, Ji Shu, Ling Qin. Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4335-4359. doi: 10.3934/dcdsb.2020100

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy