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Journal articles on the topic 'Ginzburg-Landau equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 29 January 2023

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1

CHIRON, DAVID. "BOUNDARY PROBLEMS FOR THE GINZBURG–LANDAU EQUATION." Communications in Contemporary Mathematics 07, no. 05 (October 2005): 597–648. http://dx.doi.org/10.1142/s0219199705001908.

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We provide a study at the boundary for a class of equations including the Ginzburg–Landau equation as well as the equation of travelling waves for the Gross–Pitaevskii model. We prove Clearing-Out results and an orthogonal anchoring condition of the vortex on the boundary for the Ginzburg–Landau equation with magnetic field.

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2

Adomian, G., and R. E. Meyers. "The Ginzburg-Landau equation." Computers & Mathematics with Applications 29, no. 3 (February 1995): 3–4. http://dx.doi.org/10.1016/0898-1221(94)00222-7.

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3

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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4

Li, Xiao-Yu, Yu-Lan Wang, and Zhi-Yuan Li. "Numerical simulation for the fractional-in-space Ginzburg-Landau equation using Fourier spectral method." AIMS Mathematics 8, no. 1 (2022): 2407–18. http://dx.doi.org/10.3934/math.2023124.

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<abstract><p>This paper uses the Fourier spectral method to study the propagation and interaction behavior of the fractional-in-space Ginzburg-Landau equation in different parameters and different fractional derivatives. Comparisons are made between the numerical and the exact solution, and it is found that the Fourier spectral method is a satisfactory and efficient algorithm for capturing the propagation of the fractional-in-space Ginzburg-Landau equation. Experimental findings indicate that the proposed method is easy to implement, effective and convenient in the long-time simulation for solving the proposed model. The influence of the fractional Laplacian operator on the fractional-in-space Ginzburg-Landau equation and some of the propagation behaviors of the 3D fractional-in-space Ginzburg-Landau equation are observed. In Experiment 2, we observe the propagation behaviors of the 3D fractional-in-space Ginzburg-Landau equation which are unlike any that have been previously obtained in numerical studies.</p></abstract>

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5

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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6

Barybin, Anatoly A. "Nonstationary Superconductivity: Quantum Dissipation and Time-Dependent Ginzburg-Landau Equation." Advances in Condensed Matter Physics 2011 (2011): 1–10. http://dx.doi.org/10.1155/2011/425328.

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Transport equations of the macroscopic superfluid dynamics are revised on the basis of a combination of the conventional (stationary) Ginzburg-Landau equation and Schrödinger's equation for the macroscopic wave function (often called the order parameter) by using the well-known Madelung-Feynman approach to representation of the quantum-mechanical equations in hydrodynamic form. Such an approach has given (a) three different contributions to the resulting chemical potential for the superfluid component, (b) a general hydrodynamic equation of superfluid motion, (c) the continuity equation for superfluid flow with a relaxation term involving the phenomenological parameters and , (d) a new version of the time-dependent Ginzburg-Landau equation for the modulus of the order parameter which takes into account dissipation effects and reflects the charge conservation property for the superfluid component. The conventional Ginzburg-Landau equation also follows from our continuity equation as a particular case of stationarity. All the results obtained are mutually consistent within the scope of the chosen phenomenological description and, being model-neutral, applicable to both the low- and high- superconductors.

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7

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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8

Secer, Aydin, and Yasemin Bakir. "Chebyshev wavelet collocation method for Ginzburg-Landau equation." Thermal Science 23, Suppl. 1 (2019): 57–65. http://dx.doi.org/10.2298/tsci180920330s.

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The main aim of this paper is to investigate the efficient Chebyshev wavelet collocation method for Ginzburg-Landau equation. The basic idea of this method is to have the approximation of Chebyshev wavelet series of a non-linear PDE. We demonstrate how to use the method for the numerical solution of the Ginzburg-Landau equation with initial and boundary conditions. For this purpose, we have obtained operational matrix for Chebyshev wavelets. By applying this technique in Ginzburg-Landau equation, the PDE is converted into an algebraic system of non-linear equations and this system has been solved using MAPLE computer algebra system. We demonstrate the validity and applicability of this technique which has been clarified by using an example. Exact solution is compared with an approximate solution. Moreover, Chebyshev wavelet collocation method is found to be acceptable, efficient, accurate and computational for the non-linear or PDE.

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9

Beaulieu, Anne. "Bounded solutions for an ordinary differential system from the Ginzburg–Landau theory." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 6 (August 14, 2020): 3378–408. http://dx.doi.org/10.1017/prm.2019.68.

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In this paper, we look at a linear system of ordinary differential equations as derived from the two-dimensional Ginzburg–Landau equation. In two cases, it is known that this system admits bounded solutions coming from the invariance of the Ginzburg–Landau equation by translations and rotations. The specific contribution of our work is to prove that in the other cases, the system does not admit any bounded solutions. We show that this bounded solution problem is related to an eigenvalue problem.

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10

Pascucci, Filippo, Andrea Perali, and Luca Salasnich. "Reliability of the Ginzburg–Landau Theory in the BCS-BEC Crossover by Including Gaussian Fluctuations for 3D Attractive Fermions." Condensed Matter 6, no. 4 (December 1, 2021): 49. http://dx.doi.org/10.3390/condmat6040049.

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We calculate the parameters of the Ginzburg–Landau (GL) equation of a three-dimensional attractive Fermi gas around the superfluid critical temperature. We compare different levels of approximation throughout the Bardeen–Cooper–Schrieffer (BCS) to the Bose–Einstein Condensate (BEC) regime. We show that the inclusion of Gaussian fluctuations strongly modifies the values of the Ginzburg–Landau parameters approaching the BEC regime of the crossover. We investigate the reliability of the Ginzburg–Landau theory, with fluctuations, studying the behavior of the coherence length and of the critical rotational frequencies throughout the BCS-BEC crossover. The effect of the Gaussian fluctuations gives qualitative correct trends of the considered physical quantities from the BCS regime up to the unitary limit of the BCS-BEC crossover. Approaching the BEC regime, the Ginzburg–Landau equation with the inclusion of Gaussian fluctuations turns out to be unreliable.

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11

Yang, Yisong. "On the Ginzburg-Landau Wave Equation." Bulletin of the London Mathematical Society 22, no. 2 (March 1990): 167–70. http://dx.doi.org/10.1112/blms/22.2.167.

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12

Rosier, Lionel, and Bing-Yu Zhang. "Controllability of the Ginzburg–Landau equation." Comptes Rendus Mathematique 346, no. 3-4 (February 2008): 167–72. http://dx.doi.org/10.1016/j.crma.2007.11.031.

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13

Lu, Qiuying, Guifeng Deng, and Weipeng Zhang. "Random Attractors for Stochastic Ginzburg-Landau Equation on Unbounded Domains." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/428685.

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We prove the existence of a pullback attractor inL2(ℝn)for the stochastic Ginzburg-Landau equation with additive noise on the entiren-dimensional spaceℝn. We show that the stochastic Ginzburg-Landau equation with additive noise can be recast as a random dynamical system. We demonstrate that the system possesses a uniqueD-random attractor, for which the asymptotic compactness is established by the method of uniform estimates on the tails of its solutions.

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14

Pérez Maldonado, Maximino, Haret C. Rosu, and Elizabeth Flores Garduño. "Non-autonomous Ginzburg-Landau solitons using the He-Li mapping method." CIENCIA ergo sum 27, no. 4 (November 4, 2020): e104. http://dx.doi.org/10.30878/ces.v27n4a3.

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We find and discuss the non-autonomous soliton solutions in the case of variable nonlinearity and dispersion implied by the Ginzburg-Landau equation with variable coefficients. In this work we obtain non-autonomous Ginzburg-Landau solitons from the standard autonomous Ginzburg-Landau soliton solutions using a simplified version of the He-Li mapping. We find soliton pulses of both arbitrary and fixed amplitudes in terms of a function constrained by a single condition involving the nonlinearity and the dispersion of the medium. This is important because it can be used as a tool for the parametric manipulation of these non-autonomous solitons.

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15

Wellot, Yanick Alain Servais, and Gires Dimitri Nkaya. "Analytical Solution of the Ginzburg-Landau Equation." European Journal of Pure and Applied Mathematics 15, no. 4 (October 31, 2022): 1750–59. http://dx.doi.org/10.29020/nybg.ejpam.v15i4.4551.

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In this paper, we will construct the solution of the Landau-Ginzburg equation by the Adomian decomposition method. This method avoids linearization of space and discretization of time, it often gives a good approximation of the exact solution.

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16

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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17

Staliunas, K. "Laser Ginzburg-Landau equation and laser hydrodynamics." Physical Review A 48, no. 2 (August 1, 1993): 1573–81. http://dx.doi.org/10.1103/physreva.48.1573.

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18

Ovchinnikov, Yu N., and I. M. Sigal. "The Ginzburg-Landau equation III. Vortex dynamics." Nonlinearity 11, no. 5 (September 1, 1998): 1277–94. http://dx.doi.org/10.1088/0951-7715/11/5/006.

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19

Ma, Tian, Jungho Park, and Shouhong Wang. "Dynamic Bifurcation of the Ginzburg--Landau Equation." SIAM Journal on Applied Dynamical Systems 3, no. 4 (January 2004): 620–35. http://dx.doi.org/10.1137/040603747.

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20

Pierce, R. "The Ginzburg–Landau equation for interfacial instabilities." Physics of Fluids A: Fluid Dynamics 4, no. 11 (November 1992): 2486–94. http://dx.doi.org/10.1063/1.858436.

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21

Tarasov, Vasily E., and George M. Zaslavsky. "Fractional Ginzburg–Landau equation for fractal media." Physica A: Statistical Mechanics and its Applications 354 (August 2005): 249–61. http://dx.doi.org/10.1016/j.physa.2005.02.047.

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22

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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23

Mancas, Stefan C., and Roy S. Choudhury. "Pulses and snakes in Ginzburg–Landau equation." Nonlinear Dynamics 79, no. 1 (September 20, 2014): 549–71. http://dx.doi.org/10.1007/s11071-014-1686-5.

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24

Sirovich, Lawrence, and Paul K. Newton. "Periodic solutions of the Ginzburg-Landau equation." Physica D: Nonlinear Phenomena 21, no. 1 (August 1986): 115–25. http://dx.doi.org/10.1016/0167-2789(86)90082-5.

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25

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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26

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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27

Benzi, R., A. Sutera, and A. Vulpiani. "Stochastic resonance in the Landau-Ginzburg equation." Journal of Physics A: Mathematical and General 18, no. 12 (August 21, 1985): 2239–45. http://dx.doi.org/10.1088/0305-4470/18/12/022.

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28

Bricmont, J., and A. Kupiainen. "Renormalization Group and the Ginzburg-Landau equation." Communications in Mathematical Physics 150, no. 1 (November 1992): 193–208. http://dx.doi.org/10.1007/bf02096573.

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29

Eckmann, J. P., and Th Gallay. "Front solutions for the Ginzburg-Landau equation." Communications in Mathematical Physics 152, no. 2 (March 1993): 221–48. http://dx.doi.org/10.1007/bf02098298.

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30

Lu, Kening, and Xing-Bin Pan. "Ginzburg–Landau Equation with DeGennes Boundary Condition." Journal of Differential Equations 129, no. 1 (July 1996): 136–65. http://dx.doi.org/10.1006/jdeq.1996.0114.

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31

Kurzke, Matthias, Christof Melcher, Roger Moser, and Daniel Spirn. "Ginzburg–Landau Vortices Driven by the Landau–Lifshitz–Gilbert Equation." Archive for Rational Mechanics and Analysis 199, no. 3 (September 7, 2010): 843–88. http://dx.doi.org/10.1007/s00205-010-0356-0.

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32

WANG KAI-GE. "LASER GINZBURG-LANDAU EQUATION AND ITS PHASE DIFFUSION EQUATION." Acta Physica Sinica 42, no. 2 (1993): 256. http://dx.doi.org/10.7498/aps.42.256.

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33

OVCHINNIKOV, YU N., and I. M. SIGAL. "ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF GINZBURG–LANDAU AND RELATED EQUATIONS." Reviews in Mathematical Physics 12, no. 02 (February 2000): 287–99. http://dx.doi.org/10.1142/s0129055x00000101.

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In this paper we find asymptotic behaviour of solutions of the Ginzburg–Landau equation at the spatial infinity. Our method uses very little of a particular structure of the non-linearity and can be extended to a larger class of related equations.

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34

Sameena, T., and S. Pranesh. "Synchronous and Asynchronous Boundary Temperature Modulations on Triple-Diffusive Convection in Couple Stress Liquid Using Ginzburg-Landau Model." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 645. http://dx.doi.org/10.14419/ijet.v7i4.10.21304.

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A nonlinear study of synchronous and asynchronous boundary temperature modulations on the onset of triple-diffusive convection in couple stress liquid is examined. Two cases of temperature modulations are studied: (a) Synchronous temperature modulation ( ) and (b) Asynchronous temperature modulation ( ). It is done to examine the influence of mass and heat transfer by deriving Ginzburg-Landau equation. The resultant Ginzburg-Landau equation is Bernoulli equation and it is solved numerically by means of Mathematica. The influence of solute Rayleigh numbers and couple stress parameter is studied. It is observed that couple stress parameter increases the mass and heat transfer whereas solute Rayleigh numbers decreases the mass and heat transfer.

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35

Yang, Yisong. "Global spatially periodic solutions to the Ginzburg–Landau equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 110, no. 3-4 (1988): 263–73. http://dx.doi.org/10.1017/s0308210500022253.

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36

OVCHINNIKOV, Y. N., and I. M. SIGAL. "The energy of Ginzburg–Landau vortices." European Journal of Applied Mathematics 13, no. 2 (April 2002): 153–78. http://dx.doi.org/10.1017/s0956792501004752.

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We consider the Ginzburg–Landau equation in dimension two. We introduce a key notion of the vortex (interaction) energy. It is defined by minimizing the renormalized Ginzburg–Landau (free) energy functional over functions with a given set of zeros of given local indices. We find the asymptotic behaviour of the vortex energy as the inter-vortex distances grow. The leading term of the asymptotic expansion is the vortex self-energy while the next term is the classical Kirchhoff–Onsager Hamiltonian. To derive this expansion we use several novel techniques.

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37

Yin, Lei, and Defu Hou. "s-Wave holographic superconductor in different ensembles and its thermodynamic consistency." International Journal of Modern Physics D 26, no. 04 (February 17, 2017): 1750027. http://dx.doi.org/10.1142/s0218271817500274.

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In this paper, we analytically study the consistency between the Ginzburg–Landau theory of the holographic superconductor in different ensembles and the fundamental thermodynamic relation, we derive the equation of motion of the scalar field which depicts the superconducting phase in canonical ensemble (CE) and a consistent formula to connect the holographic order-parameter to the Ginzburg–Landau coefficients in different thermodynamic ensembles, and we also study the spatially nonuniform Helmholtz free energy.

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38

Dias, João-Paulo, Filipe Oliveira, and Hugo Tavares. "On a coupled system of a Ginzburg–Landau equation with a quasilinear conservation law." Communications in Contemporary Mathematics 22, no. 07 (June 13, 2019): 1950054. http://dx.doi.org/10.1142/s0219199719500548.

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We study a coupled system of a complex Ginzburg–Landau equation with a quasilinear conservation law [Formula: see text] which can describe the interaction between a laser beam and a fluid flow (see [I.-S. Aranson and L. Kramer, The world of the complex Ginzburg–Landau equation, Rev. Mod. Phys. 74 (2002) 99–143]). We prove the existence of a local in time strong solution for the associated Cauchy problem and, for a certain class of flux functions, the existence of global weak solutions. Furthermore, we prove the existence of standing wave solutions of the form [Formula: see text] in several cases.

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39

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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40

TAGARE, S. G. "Nonlinear stationary magnetoconvection in a rotating fluid." Journal of Plasma Physics 58, no. 3 (October 1997): 395–408. http://dx.doi.org/10.1017/s0022377897006004.

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We investigate finite-amplitude magnetoconvection in a rotating fluid in the presence of a vertical magnetic field when the axis of rotation is parallel to a vertical magnetic field. We derive a nonlinear, time-dependent, one-dimensional Landau–Ginzburg equation near the onset of stationary convection at supercritical pitchfork bifurcation whenformula hereand a nonlinear time-dependent second-order ordinary differential equation when Ta=T*a (from below). Ta=T*a corresponds to codimension-two bifurcation (or secondary bifurcation), where the threshold for stationary convection at the pitchfork bifurcation coincides with the threshold for oscillatory convection at the Hopf bifurcation. We obtain steady-state solutions of the one-dimensional Landau–Ginzburg equation, and discuss the solution of the nonlinear time-dependent second-order ordinary differential equation.

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41

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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42

Kiran, Palle, B. S. Bhadauria, and R. Roslan. "The Effect of Throughflow on Weakly Nonlinear Convection in a Viscoelastic Saturated Porous Medium." Journal of Nanofluids 9, no. 1 (March 1, 2020): 36–46. http://dx.doi.org/10.1166/jon.2020.1724.

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In this paper we have investigated the effect of throughflow on thermal convection in a viscoelastic fluid saturated porous media. The governing equations are modelled in the presence of throughflow. These equations are made dimensionless and the obtained nonlinear problem solved numerically. There are two types of throughflow effects on thermal instability inflow and outflow investigated by finite amplitude analysis. This finite amplitude equation is obtained using the complex Ginzburg-Landau amplitude equation (CGLE) for a weak nonlinear oscillatory convection. The heat transport analysis is given by complex Ginzburg-Landau amplitude equation (CGLE). The numerical results indicate that due to the non-uniform throughflow there is instability at the bottom plate and influence the heat transfer in the system. The vertical throughflow is having both stable and unstable modes depending on flow direction. The nature of viscoelastic fluid is having both effects either stabilize or destabilize. Further, it is found that the nonlinear throughflow effects have dual role on heat transport. The solutions of the present problem are obtained numerically by using Runge-Kutta fourth order method.

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43

Ma, Li. "Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation." International Journal of Mathematics 27, no. 05 (May 2016): 1650048. http://dx.doi.org/10.1142/s0129167x16500488.

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In this paper, we give the boundedness of solutions to Ginzburg–Landau fractional Laplacian equation, which extends the Herve–Herve theorem into the nonlinear fractional Laplacian equation. We follow Brezis’ idea to use the Kato inequality. A related linear fractional Schrödinger equation is also studied.

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44

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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45

İNÇ, M., and N. BİLDİK. "Non - Perturbative Solution of the Ginzburg - Landau Equation." Mathematical and Computational Applications 5, no. 2 (August 1, 2000): 113–17. http://dx.doi.org/10.3390/mca5020113.

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46

Bethuel, F., and G. Orlandi. "Uniform estimates for the parabolic Ginzburg–Landau equation." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 219–38. http://dx.doi.org/10.1051/cocv:2002026.

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47

Kengne, E. "Modified Ginzburg–Landau Equation and Benjamin–Feir Instability." Nonlinear Oscillations 6, no. 3 (July 2003): 339–49. http://dx.doi.org/10.1023/b:nono.0000016412.08459.72.

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48

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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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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Gu, Xian-Ming, Lin Shi, and Tianhua Liu. "Well-posedness of the fractional Ginzburg–Landau equation." Applicable Analysis 98, no. 14 (May 30, 2018): 2545–58. http://dx.doi.org/10.1080/00036811.2018.1466281.

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