Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations

Kolade M. Owolabi; Edson Pindza

doi:10.3934/dcdss.2020048

Article Contents

2020, Volume 13, Issue 3: 835-851. Doi: 10.3934/dcdss.2020048

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Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations

Kolade M. Owolabi^1, , and
Edson Pindza^2,3,

1.
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2.
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 002, South Africa
3.
Achieversklub School of Cryptocurrency and Entrepreneurship, 1 Sturdee Avenue, Rosebank 2196, South Africa

^* Corresponding author: koladematthewowolabi@tdtu.edu.vn (K. M. Owolabi)
^* Corresponding author: koladematthewowolabi@tdtu.edu.vn (K. M. Owolabi)

Received: April 30, 2018

Revised: July 31, 2018

Published: December 2019

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Abstract

Ginzburg-Landau equation has a rich record of success in describing a vast variety of nonlinear phenomena such as liquid crystals, superfluidity, Bose-Einstein condensation and superconductivity to mention a few. Fractional order equations provide an interesting bridge between the diffusion wave equation of mathematical physics and intuition generation, it is of interest to see if a similar generalization to fractional order can be useful here. Non-integer order partial differential equations describing the chaotic and spatiotemporal patterning of fractional Ginzburg-Landau problems, mostly defined on simple geometries like triangular domains, are considered in this paper. We realized through numerical experiments that the Ginzburg-Landau equation world is bounded between the limits where new phenomena and scenarios evolve, such as sink and source solutions (spiral patterns in 2D and filament-like structures in 3D), various core and wave instabilities, absolute instability versus nonlinear convective cases, competition and interaction between sources and chaos spatiotemporal states. For the numerical simulation of these kind of problems, spectral methods provide a fast and efficient approach.

Keywords:

Mathematics Subject Classification: Primary: 34A34, 35A05, 35K57; Secondary: 65L05, 65M06, 93C10.

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Figure 1. Space-time mesh results of (24) showing chaotic states in the spatiotemporal regime for parameters $ (b,c) = (1, -1.3) $ at different instances of fractional index $ \alpha $ and $ t = 40 $. simulation runs for $ N = 200 $ with step size $ h = 0.1 $

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Figure 2. Solution of the 2D fractional complex Ginzburg-Landau equation on $ [0,200]\times [0,200] $ with $ b = 1 $ for both the focusing case: $ c = 1.3 $ (first-column) and the defocussing case: $ c = -1.3 $ (second-column) at final time $ t = 100 $, $ \alpha = (0.85, 1.0, 1.50) $ and $ N = 200 $

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Figure 3. The first and second columns represent 3D results of the fractional complex Ginzburg-Landau equation on $ [0, 20]^3 $ obtained at instances $ \alpha = (0.5, 1.0, 1.50) $ for random and initial conditions respectively. Other parameters are: $ b = 1, L = 20 $ and final time $ t = 10 $ (N = 100)

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Figure 4. The 2D results of fractional-in-space problem (24) showing the bound state of oppositely- and like-charged spirals at some instances of fractional power $ \alpha $. simulation runs for $ N = 200 $

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Figure 5. The 3D isosurfaces of $ |u(x,y,z)| $ of (24) showing chaotic patterns at different instances of $ \alpha $ for $ \epsilon = -0.05; b = 1.0, \phi = 1.0, c = 1.3, \psi = 1.0, d = 0.105, \varphi = 0.03, L = 20 $ and final time $ t = 20 $. Simulation runs for $ N = 64 $

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