# American Institute of Mathematical Sciences

## Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations

 1 Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa 2 Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria 3 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 002, South Africa 4 Achieversklub School of Cryptocurrency and Entrepreneurship, 1 Sturdee Avenue, Rosebank 2196, South Africa

* Corresponding author: pindzaedson@yahoo.fr (K. M. Owolabi)

Received  May 2018 Revised  August 2018 Published  March 2019

Fund Project: The research contained in this report is supported by South African National Research Foundation

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.

Citation: Kolade M. Owolabi, Edson Pindza. Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020048
##### References:
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Kay and K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT Numerical Mathematics, 54 (2014), 937-954. doi: 10.1007/s10543-014-0484-2. Google Scholar [13] M. Caputo, Linear models of dissipation whose $\mathcal{Q}$ is almost frequency independent: Part Ⅱ, Geophysical Journal International, 13 (1967), 529-539: Reprinted in: Fractional Calculus and Applied Analysis, 11 (2008), 3-14. Google Scholar [14] A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Vienna-New York, 1997.Google Scholar [15] C. Cartes, J. Cisternas, O. Descalzi and H. R. Brand, Model of a two-dimensional extended chaotic system: Evidence of diffusing dissipative solitons, Physical Review Letters, 109 (2012), 178303. doi: 10.1103/PhysRevLett.109.178303. Google Scholar [16] S. M. Cox and P. C. 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Turner, A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation, Applied Mathematics and Computation, 256 (2015), 930-938. doi: 10.1016/j.amc.2015.01.092. Google Scholar [26] Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnamica, 24 (1999), 207-233. Google Scholar [27] C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Analysis, 68 (2008), 461-484. doi: 10.1016/j.na.2006.11.011. Google Scholar [28] S. Momani and Z. Odibat, A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor's Formula, Journal of Computational and Applied Mathematics, 220 (2008), 85-95. doi: 10.1016/j.cam.2007.07.033. Google Scholar [29] H. Munthe-Kaas, High order Runge-Kutta methods on manifolds, Applied Numerical Mathematics, 29 (1999), 115-127. doi: 10.1016/S0168-9274(98)00030-0. Google Scholar [30] A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, Journal of Fluid Mechanics, 38 (1969), 279-303. doi: 10.1017/S0022112069000176. Google Scholar [31] Z. Odibat and S. Momani, Numerical methods for nonlinear partial differential equations of fractional order, Applied Mathematical Modeling, 32 (2008), 28-39. doi: 10.1016/j.apm.2006.10.025. Google Scholar [32] K. M. Owolabi, Robust IMEX schemes for solving two-dimensional reaction-diffusion models, International Journal of Nonlinear Science and Numerical Simulations, 16 (2015), 271-284. doi: 10.1515/ijnsns-2015-0004. Google Scholar [33] K. M. Owolabi and K. C. Patidar, Existence and permanence in a diffusive KiSS model with robust numerical simulations, International Journal of Differential Equations, 2015 (2015), Art. ID 485860, 8 pp. doi: 10.1155/2015/485860. Google Scholar [34] K. M. Owolabi and K. C. Patidar, Numerical simulations of multicomponent ecological models with adaptive methods, Theoretical Biology and Medical Modelling, 13 (2016), p1. doi: 10.1186/s12976-016-0027-4. Google Scholar [35] K. M. Owolabi and K. C. Patidar, Effect of spatial configuration of an extended nonlinear Kierstead Slobodkin reaction transport model with adaptive numerical scheme, Springer Plus, 5 (2016), 303. doi: 10.1186/s40064-016-1941-y. Google Scholar [36] K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulations, 44 (2017), 304-317. doi: 10.1016/j.cnsns.2016.08.021. Google Scholar [37] K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos, Solitons and Fractals, 103 (2017), 544-554. doi: 10.1016/j.chaos.2017.07.013. Google Scholar [38] K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 7, 17 pp. doi: 10.1051/mmnp/2018006. Google Scholar [39] K. M. Owolabi and A. Atangana, Modelling and formation of spatiotemporal patterns of fractional predation system in subdiffusion and superdiffusion scenarios, The European physical Journal Plus, 133 (2018), 43. doi: 10.1140/epjp/i2018-11886-2. Google Scholar [40] K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, The European physical Journal Plus, 133 (2018), 15. doi: 10.1140/epjp/i2018-11863-9. Google Scholar [41] K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, The European Physical Journal Plus, 133 (2018), 98. doi: 10.1140/epjp/i2018-11951-x. Google Scholar [42] K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos, Solitons and Fractals, 111 (2018), 119-127. doi: 10.1016/j.chaos.2018.04.019. Google Scholar [43] E. Pindza and K. M. Owolabi, Fourier spectral method for higher order space fractional reaction-diffusion equations, Communications in Nonlinear Science and Numerical Simulation, 40 (2016), 112-128. doi: 10.1016/j.cnsns.2016.04.020. Google Scholar [44] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. Google Scholar [45] X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Applicable Analysis, 92 (2013), 318-334. doi: 10.1080/00036811.2011.614601. Google Scholar [46] Yu. A. Rossikhin and M. V. Shitikova., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997), 15-67. doi: 10.1115/1.3101682. Google Scholar [47] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, New York, 1993. Google Scholar [48] E. Sousa, Numerical approximations for fractional diffusion equations via splines, Computers and Mathematics with Applications, 62 (2011), 938-944. doi: 10.1016/j.camwa.2011.04.015. Google Scholar [49] K. Stewartson and J. T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, Journal of Fluid Mechanics, 48 (1971), 529-545. doi: 10.1017/S0022112071001733. Google Scholar [50] J. Wu, Generalized MHD equations, Journal of Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007. Google Scholar [51] F. Zeng, F. Liu, C. Li, K. Burrage, I. Turner and V. Anh, A Crank-Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation, SIAM Journal on Numerical Analysis, 52 (2014), 2599-2622. doi: 10.1137/130934192. Google Scholar [52] F. Zeng, C. Li, F. Liu and I. Turner, Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy, SIAM Journal on Scientific Computing, 37 (2015), A55–A78. doi: 10.1137/14096390X. Google Scholar [53] Z. Zhai, Well-posedness for fractional Navier-Stokes equations in critical spaces close to $B_{\infty, \infty}^{-(2\beta-1)(R^n)}$, Dynamics of PDE, 7 (2010), 25-44. doi: 10.4310/DPDE.2010.v7.n1.a2. Google Scholar [54] Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Annales de l'Institut Henri Poincaré, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014. Google Scholar [55] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, New Jersey, 2014. doi: 10.1142/9069. Google Scholar

show all references

##### References:
 [1] G. Akrivis, Implicit-explicit multistep methods for nonlinear parabolic equations, Mathematical Computation, 82 (2013), 45-68. doi: 10.1090/S0025-5718-2012-02628-7. Google Scholar [2] G. Akrivis, Stability properties of implicit-explicit multistep methods for a class of nonlinear parabolic equations, Mathematical Computation, 85 (2016), 2217-2229. doi: 10.1090/mcom/3070. Google Scholar [3] A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, Journal of Computational Physics, 293 (2015), 104-114. doi: 10.1016/j.jcp.2014.12.043. Google Scholar [4] A. Atangana, Derivative with a New Parameter : Theory, Methods and Applications, Academic Press, New York, 2016. doi: 10.1016/B978-0-08-100644-3.00001-5. Google Scholar [5] A. Atangana, Fractional Operators With Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018. Google Scholar [6] A. Atangana and S. Jain, A new numerical approximation of the fractal ordinary differential equation, The European Physical Journal Plus, 133 (2018), 37. doi: 10.1140/epjp/i2018-11895-1. Google Scholar [7] A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 3, 21 pp. doi: 10.1051/mmnp/2018010. Google Scholar [8] F. Bérad and S. C. Mancas, Spatiotemporal two-dimensional solitons in the complex Ginzburg-Landau equations, Advances and Applications in Fluid Mechanics, 8 (2011), 141-156. Google Scholar [9] P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations, Journal of Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458. Google Scholar [10] G. W. S. Blair, The role of psychophysics in rheology, Journal Colloid Sciences, 2 (1947), 21-32. Google Scholar [11] G. W. S. Blair, Measurements of Mind and Matter, Dennis Dobson, London, 1950.Google Scholar [12] A. Bueno-Orovio, D. Kay and K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT Numerical Mathematics, 54 (2014), 937-954. doi: 10.1007/s10543-014-0484-2. Google Scholar [13] M. Caputo, Linear models of dissipation whose $\mathcal{Q}$ is almost frequency independent: Part Ⅱ, Geophysical Journal International, 13 (1967), 529-539: Reprinted in: Fractional Calculus and Applied Analysis, 11 (2008), 3-14. Google Scholar [14] A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Vienna-New York, 1997.Google Scholar [15] C. Cartes, J. Cisternas, O. Descalzi and H. R. Brand, Model of a two-dimensional extended chaotic system: Evidence of diffusing dissipative solitons, Physical Review Letters, 109 (2012), 178303. doi: 10.1103/PhysRevLett.109.178303. Google Scholar [16] S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002), 430-455. doi: 10.1006/jcph.2002.6995. Google Scholar [17] M. Davison and C. Essex, Fractional differential equations and initial value problems, Mathematical Scientist, 23 (1998), 108-116. Google Scholar [18] V. Garca-Morales and K. Krischer, The complex Ginzburg-Landau equation: An introduction, Contemporary Physics, 53 (2012), 79-95. doi: 10.1080/00107514.2011.642554. Google Scholar [19] A. N. Gerasimov, A generalization of linear laws of deformation and its application to inner friction problems, Prikl. Mat. Mekh., 12 (1948), 251-259. Google Scholar [20] B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, World Scientific, Singapore, 2015. doi: 10.1142/9543. Google Scholar [21] S. Jain, Numerical analysis for the fractional diffusion and fractional Buckmaster's equation by two step adam- bashforth method, The European Physical Journal Plus, 133 (2018), 19.Google Scholar [22] A. K. Kassam and L. N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM Journal on Scientific Computing, 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633. Google Scholar [23] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Netherlands, 2006. Google Scholar [24] F. Liu, P. Zhang, V. Anh and I. Turner, A fractional-order implicit difference approximation for the space-time fractional diffusion equation, ANZIAM J., 47 (2006), 48-68. doi: 10.1017/S1446181100009998. Google Scholar [25] Q. Liu, F. Liu, Y. Gu, P. Zhuang, J. Chen and I. Turner, A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation, Applied Mathematics and Computation, 256 (2015), 930-938. doi: 10.1016/j.amc.2015.01.092. Google Scholar [26] Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnamica, 24 (1999), 207-233. Google Scholar [27] C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Analysis, 68 (2008), 461-484. doi: 10.1016/j.na.2006.11.011. Google Scholar [28] S. Momani and Z. Odibat, A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor's Formula, Journal of Computational and Applied Mathematics, 220 (2008), 85-95. doi: 10.1016/j.cam.2007.07.033. Google Scholar [29] H. Munthe-Kaas, High order Runge-Kutta methods on manifolds, Applied Numerical Mathematics, 29 (1999), 115-127. doi: 10.1016/S0168-9274(98)00030-0. Google Scholar [30] A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, Journal of Fluid Mechanics, 38 (1969), 279-303. doi: 10.1017/S0022112069000176. Google Scholar [31] Z. Odibat and S. Momani, Numerical methods for nonlinear partial differential equations of fractional order, Applied Mathematical Modeling, 32 (2008), 28-39. doi: 10.1016/j.apm.2006.10.025. Google Scholar [32] K. M. Owolabi, Robust IMEX schemes for solving two-dimensional reaction-diffusion models, International Journal of Nonlinear Science and Numerical Simulations, 16 (2015), 271-284. doi: 10.1515/ijnsns-2015-0004. Google Scholar [33] K. M. Owolabi and K. C. Patidar, Existence and permanence in a diffusive KiSS model with robust numerical simulations, International Journal of Differential Equations, 2015 (2015), Art. ID 485860, 8 pp. doi: 10.1155/2015/485860. Google Scholar [34] K. M. Owolabi and K. C. Patidar, Numerical simulations of multicomponent ecological models with adaptive methods, Theoretical Biology and Medical Modelling, 13 (2016), p1. doi: 10.1186/s12976-016-0027-4. Google Scholar [35] K. M. Owolabi and K. C. Patidar, Effect of spatial configuration of an extended nonlinear Kierstead Slobodkin reaction transport model with adaptive numerical scheme, Springer Plus, 5 (2016), 303. doi: 10.1186/s40064-016-1941-y. Google Scholar [36] K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulations, 44 (2017), 304-317. doi: 10.1016/j.cnsns.2016.08.021. Google Scholar [37] K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos, Solitons and Fractals, 103 (2017), 544-554. doi: 10.1016/j.chaos.2017.07.013. Google Scholar [38] K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 7, 17 pp. doi: 10.1051/mmnp/2018006. Google Scholar [39] K. M. Owolabi and A. Atangana, Modelling and formation of spatiotemporal patterns of fractional predation system in subdiffusion and superdiffusion scenarios, The European physical Journal Plus, 133 (2018), 43. doi: 10.1140/epjp/i2018-11886-2. Google Scholar [40] K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, The European physical Journal Plus, 133 (2018), 15. doi: 10.1140/epjp/i2018-11863-9. Google Scholar [41] K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, The European Physical Journal Plus, 133 (2018), 98. doi: 10.1140/epjp/i2018-11951-x. Google Scholar [42] K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos, Solitons and Fractals, 111 (2018), 119-127. doi: 10.1016/j.chaos.2018.04.019. Google Scholar [43] E. Pindza and K. M. Owolabi, Fourier spectral method for higher order space fractional reaction-diffusion equations, Communications in Nonlinear Science and Numerical Simulation, 40 (2016), 112-128. doi: 10.1016/j.cnsns.2016.04.020. Google Scholar [44] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. Google Scholar [45] X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Applicable Analysis, 92 (2013), 318-334. doi: 10.1080/00036811.2011.614601. Google Scholar [46] Yu. A. Rossikhin and M. V. Shitikova., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997), 15-67. doi: 10.1115/1.3101682. Google Scholar [47] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, New York, 1993. Google Scholar [48] E. Sousa, Numerical approximations for fractional diffusion equations via splines, Computers and Mathematics with Applications, 62 (2011), 938-944. doi: 10.1016/j.camwa.2011.04.015. Google Scholar [49] K. Stewartson and J. T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, Journal of Fluid Mechanics, 48 (1971), 529-545. doi: 10.1017/S0022112071001733. Google Scholar [50] J. Wu, Generalized MHD equations, Journal of Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007. Google Scholar [51] F. Zeng, F. Liu, C. Li, K. Burrage, I. Turner and V. Anh, A Crank-Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation, SIAM Journal on Numerical Analysis, 52 (2014), 2599-2622. doi: 10.1137/130934192. Google Scholar [52] F. Zeng, C. Li, F. Liu and I. Turner, Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy, SIAM Journal on Scientific Computing, 37 (2015), A55–A78. doi: 10.1137/14096390X. Google Scholar [53] Z. Zhai, Well-posedness for fractional Navier-Stokes equations in critical spaces close to $B_{\infty, \infty}^{-(2\beta-1)(R^n)}$, Dynamics of PDE, 7 (2010), 25-44. doi: 10.4310/DPDE.2010.v7.n1.a2. Google Scholar [54] Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Annales de l'Institut Henri Poincaré, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014. Google Scholar [55] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, New Jersey, 2014. doi: 10.1142/9069. Google Scholar
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 $h = 0.1$
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 $N = 200$
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)
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 $N = 200$
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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