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May  2017, 37(5): 2539-2564. doi: 10.3934/dcds.2017109

Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation

1. 

School of Mathematics and Statistics, Shandong University, Weihai 264209, China

2. 

School of Mathematics and Systems Science, Beihang University, Beijing 100191, China

3. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

Received  July 2015 Revised  December 2016 Published  February 2017

We study the fractional complex Ginzburg-Landau equation with periodic initial boundary value condition in three spatial dimensions. The problem is discretized fully by Fourier Galerkin spectral method. The dynamical behavior of the resulting discrete system is examined. The existence of a global attractor is established, and the corresponding convergence is proved through the error estimates of the discrete solution. Numerical stability and convergence of the discrete scheme are proved.

Citation: Hong Lu, Shujuan Lü, Mingji Zhang. Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2539-2564. doi: 10.3934/dcds.2017109
References:
[1]

C. BjorlandL. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380.

[2]

K. Bogdan and T. Byczkowski, On the Schrödinger operator based on the fractional Laplacian, Bull. Polish Acad. Sci. Math., 49 (2001), 291-301.

[3]

T. Bojdecki and L. G. Gorostiza, Fractional Brownian motion via fractional Laplacian, Statist. Probab. Lett., 44 (1999), 107-108. doi: 10.1016/S0167-7152(99)00014-0.

[4]

C. Bu, On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation, J. Aust. Math. Soc. Ser. B, 36 (1995), 313-324. doi: 10.1017/S0334270000010468.

[5]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[6]

L. CaffarelliS. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.

[7]

C. Canuto and A. Qarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38 (1982), 201-229. doi: 10.1090/S0025-5718-1982-0637287-3.

[8]

C. R. DoeringJ. D. GibbonD. Holm and B. Nicolaenko, Low-dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity, 1 (1988), 279-309.

[9]

C. R. DoeringJ. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Physica D, 71 (1994), 258-318. doi: 10.1016/0167-2789(94)90150-3.

[10]

J. Dong and M. Xu, Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017. doi: 10.1016/j.jmaa.2008.03.061.

[11]

Z. E. A. Fellah, C. Depollier and M. Fellah, Propagation of ultrasonic pulses in porous elastic solids: A time domain analysis with fractional derivatives, 5-th International Conference on Mathematical and Numerical Aspects of Wave Propagation. Santiago de Compostela, Spain, 2000.

[12]

R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $ \mathbb{R}^n$, Acta Math., 210 (2013), 261-318.

[13]

V. L. Ginzburg, Nobel lecture: On superconductivity and superfluidity (what I have and have not managed to do) as well as on the "physical minimum" at the beginning of the XXI century, Reviews of Modern Physics, 76 (2004), 981-998. doi: 10.1103/RevModPhys.76.981.

[14]

V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, On Superconductivity and Superfluidity, (2009), 113-137. doi: 10.1007/978-3-540-68008-6_4.

[15]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differential Equations, 36 (2010), 247-255. doi: 10.1080/03605302.2010.503769.

[16]

B. GuoY. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. and Comput., 204 (2008), 468-477. doi: 10.1016/j.amc.2008.07.003.

[17]

B. Guo and M. Zeng, Solutions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl., 361 (2010), 131-138. doi: 10.1016/j.jmaa.2009.09.009.

[18]

H. LuP. BatesS. Lü and M. Zhang, Dynamics of the 3-D fractional complex Ginzburg-Landau equation, J. Differential Equation, 259 (2015), 5276-5301. doi: 10.1016/j.jde.2015.06.028.

[19]

H. LuP. BatesS. Lü and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. in Math. Sci., 14 (2016), 273-295. doi: 10.4310/CMS.2016.v14.n1.a11.

[20]

H. LuS. Lü and Z. Feng, Asymptotic dynamics of 2D fractional complex Ginzburg-Landau equation, I. J. Bifurcation and Chaos, 23 (2013), 1350202, 12pp. doi: 10.1142/S0218127413502027.

[21]

S. Lü and Q. Lu, Asymptotic behavior of three-dimensional Ginzburg-Landau type equation, Dynamics of Continuous, Discrete and Impulsive Systems, Series A, 13 (2006), 209-220.

[22]

S. Lü and Q. Lu, Exponential attractor for the 3D Ginzburg-Landau type equation, Nonlinear Analysis, 67 (2007), 3116-3135.

[23]

S. Lü, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation, Numer. Math., 22 (2000), 1-9.

[24]

V. G. Mazja, Sobolev Spaces, Springer-Verlag, 1985. doi: 10.1007/978-3-662-09922-3.

[25]

E. W. Montroll and M. F. Shlesinger, On the wonderful world of random walks, in: J. Leibowitz and E. W. Montroll (Eds. ), Nonequilibrium Phenomena Ⅱ: from Stochastics to Hydrodynamics, North-Holland, Amsterdam, 1984, 1-121

[26]

R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. stat. solidi. B, 133 (1986), 425-430. doi: 10.1002/pssb.2221330150.

[27]

K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation, Physica D, 41 (1990), 232-252. doi: 10.1016/0167-2789(90)90125-9.

[28]

X. Pu and B. Guo, Global weak solutions of the fractional Landau-Lifshitz-Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98. doi: 10.1016/j.jmaa.2010.06.035.

[29]

A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: Solutions and applications, Chaos, 7 (1997), 753-764. doi: 10.1063/1.166272.

[30]

S. Salsa, Optimal regularity in lower dimensional obstacle problems. Subelliptic PDE's and applications to geometry and finance, Lect. Notes Semin. Interdiscip. Mat. , 6, Semin. Interdiscip. Mat. (S. I. M. ), Potenza, 2007,217-226

[31]

J. Shen, Long-time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-209. doi: 10.1080/00036819008839963.

[32]

M. F. ShlesingerG. M. Zaslavsky and J. Klafter, Strange Kinetics, Nature, 363 (1993), 31-37. doi: 10.1038/363031a0.

[33]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.

[34]

R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1995.

[35]

V. E. Tarasov and G. M. Zaslavsky, Fractional Ginzburg-Landau equation for fractal media, Physica A, 354 (2005), 249-261. doi: 10.1016/j.physa.2005.02.047.

[36]

V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16 (2006), 023110.

[37]

H. Weitzner and G. M. Zaslavsky, Some applications of fractional derivatives, Communications in Nonlinear Science and Numerical Simulation, 8 (2003), 273-281.

[38]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371 (2002), 461-580. doi: 10.1016/S0370-1573(02)00331-9.

[39] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, University Press, Oxford, 2005.
[40]

G. M. Zaslavsky and M. Edelman, Weak mixing and anomalous kinetics along filamented surfaces, Chaos, 11 (2001), 295-305. doi: 10.1063/1.1355358.

show all references

References:
[1]

C. BjorlandL. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380.

[2]

K. Bogdan and T. Byczkowski, On the Schrödinger operator based on the fractional Laplacian, Bull. Polish Acad. Sci. Math., 49 (2001), 291-301.

[3]

T. Bojdecki and L. G. Gorostiza, Fractional Brownian motion via fractional Laplacian, Statist. Probab. Lett., 44 (1999), 107-108. doi: 10.1016/S0167-7152(99)00014-0.

[4]

C. Bu, On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation, J. Aust. Math. Soc. Ser. B, 36 (1995), 313-324. doi: 10.1017/S0334270000010468.

[5]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[6]

L. CaffarelliS. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.

[7]

C. Canuto and A. Qarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38 (1982), 201-229. doi: 10.1090/S0025-5718-1982-0637287-3.

[8]

C. R. DoeringJ. D. GibbonD. Holm and B. Nicolaenko, Low-dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity, 1 (1988), 279-309.

[9]

C. R. DoeringJ. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Physica D, 71 (1994), 258-318. doi: 10.1016/0167-2789(94)90150-3.

[10]

J. Dong and M. Xu, Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017. doi: 10.1016/j.jmaa.2008.03.061.

[11]

Z. E. A. Fellah, C. Depollier and M. Fellah, Propagation of ultrasonic pulses in porous elastic solids: A time domain analysis with fractional derivatives, 5-th International Conference on Mathematical and Numerical Aspects of Wave Propagation. Santiago de Compostela, Spain, 2000.

[12]

R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $ \mathbb{R}^n$, Acta Math., 210 (2013), 261-318.

[13]

V. L. Ginzburg, Nobel lecture: On superconductivity and superfluidity (what I have and have not managed to do) as well as on the "physical minimum" at the beginning of the XXI century, Reviews of Modern Physics, 76 (2004), 981-998. doi: 10.1103/RevModPhys.76.981.

[14]

V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, On Superconductivity and Superfluidity, (2009), 113-137. doi: 10.1007/978-3-540-68008-6_4.

[15]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differential Equations, 36 (2010), 247-255. doi: 10.1080/03605302.2010.503769.

[16]

B. GuoY. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. and Comput., 204 (2008), 468-477. doi: 10.1016/j.amc.2008.07.003.

[17]

B. Guo and M. Zeng, Solutions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl., 361 (2010), 131-138. doi: 10.1016/j.jmaa.2009.09.009.

[18]

H. LuP. BatesS. Lü and M. Zhang, Dynamics of the 3-D fractional complex Ginzburg-Landau equation, J. Differential Equation, 259 (2015), 5276-5301. doi: 10.1016/j.jde.2015.06.028.

[19]

H. LuP. BatesS. Lü and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. in Math. Sci., 14 (2016), 273-295. doi: 10.4310/CMS.2016.v14.n1.a11.

[20]

H. LuS. Lü and Z. Feng, Asymptotic dynamics of 2D fractional complex Ginzburg-Landau equation, I. J. Bifurcation and Chaos, 23 (2013), 1350202, 12pp. doi: 10.1142/S0218127413502027.

[21]

S. Lü and Q. Lu, Asymptotic behavior of three-dimensional Ginzburg-Landau type equation, Dynamics of Continuous, Discrete and Impulsive Systems, Series A, 13 (2006), 209-220.

[22]

S. Lü and Q. Lu, Exponential attractor for the 3D Ginzburg-Landau type equation, Nonlinear Analysis, 67 (2007), 3116-3135.

[23]

S. Lü, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation, Numer. Math., 22 (2000), 1-9.

[24]

V. G. Mazja, Sobolev Spaces, Springer-Verlag, 1985. doi: 10.1007/978-3-662-09922-3.

[25]

E. W. Montroll and M. F. Shlesinger, On the wonderful world of random walks, in: J. Leibowitz and E. W. Montroll (Eds. ), Nonequilibrium Phenomena Ⅱ: from Stochastics to Hydrodynamics, North-Holland, Amsterdam, 1984, 1-121

[26]

R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. stat. solidi. B, 133 (1986), 425-430. doi: 10.1002/pssb.2221330150.

[27]

K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation, Physica D, 41 (1990), 232-252. doi: 10.1016/0167-2789(90)90125-9.

[28]

X. Pu and B. Guo, Global weak solutions of the fractional Landau-Lifshitz-Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98. doi: 10.1016/j.jmaa.2010.06.035.

[29]

A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: Solutions and applications, Chaos, 7 (1997), 753-764. doi: 10.1063/1.166272.

[30]

S. Salsa, Optimal regularity in lower dimensional obstacle problems. Subelliptic PDE's and applications to geometry and finance, Lect. Notes Semin. Interdiscip. Mat. , 6, Semin. Interdiscip. Mat. (S. I. M. ), Potenza, 2007,217-226

[31]

J. Shen, Long-time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-209. doi: 10.1080/00036819008839963.

[32]

M. F. ShlesingerG. M. Zaslavsky and J. Klafter, Strange Kinetics, Nature, 363 (1993), 31-37. doi: 10.1038/363031a0.

[33]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.

[34]

R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1995.

[35]

V. E. Tarasov and G. M. Zaslavsky, Fractional Ginzburg-Landau equation for fractal media, Physica A, 354 (2005), 249-261. doi: 10.1016/j.physa.2005.02.047.

[36]

V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16 (2006), 023110.

[37]

H. Weitzner and G. M. Zaslavsky, Some applications of fractional derivatives, Communications in Nonlinear Science and Numerical Simulation, 8 (2003), 273-281.

[38]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371 (2002), 461-580. doi: 10.1016/S0370-1573(02)00331-9.

[39] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, University Press, Oxford, 2005.
[40]

G. M. Zaslavsky and M. Edelman, Weak mixing and anomalous kinetics along filamented surfaces, Chaos, 11 (2001), 295-305. doi: 10.1063/1.1355358.

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