March  2016, 21(2): 575-590. doi: 10.3934/dcdsb.2016.21.575

Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise

1. 

School of Mathematics and Systems Science & LMIB, Beijing University of Aeronautics and Astronautics, Beijing, 100191

2. 

School of Mathematics and Systems Science, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

3. 

School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539, United States

Received  December 2014 Revised  May 2015 Published  November 2015

In this work, we analyze the stochastic fractional Ginzburg-Landau equation with multiplicative noise in two spatial dimensions with a particular interest in the asymptotic behavior of its solutions. To get started, we first transfer the stochastic fractional Ginzburg-Landau equation into a random equation whose solutions generate a random dynamical system. The existence of a random attractor for the resulting random dynamical system is explored, and the Hausdorff dimension of the random attractor is estimated.
Citation: Shujuan Lü, Hong Lu, Zhaosheng Feng. Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 575-590. doi: 10.3934/dcdsb.2016.21.575
References:
[1]

H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynamics and Differential Equations, 9 (1997), 307. doi: 10.1007/BF02219225. Google Scholar

[2]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar

[3]

A. Debussche, Hausdorff dimension of a random invariant set,, J. Math. Pure Appl., 77 (1998), 967. doi: 10.1016/S0021-7824(99)80001-4. Google Scholar

[4]

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

[5]

C. W. Gardiner, Handbooks of Stochastic Methods for Physics,, Chemistry and Natural Sciences, (1983). doi: 10.1007/978-3-662-02377-8. Google Scholar

[6]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation,, Commun. Partial Differential Equations, 36 (2011), 247. doi: 10.1080/03605302.2010.503769. Google Scholar

[7]

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

[8]

S. Holm and S. P. Näsholm, Comparison of fractional wave equations for power law attenuation in ultrasound and elastography,, Ultrasound Med. Biol., 40 (2014), 695. doi: 10.1016/j.ultrasmedbio.2013.09.033. Google Scholar

[9]

N. Laskin, Fractional Schrödinger equation,, Physical Review E, 66 (2002). doi: 10.1103/PhysRevE.66.056108. Google Scholar

[10]

H. Lu, S. Lü and Z. Feng, Asymptotic dynamics of 2d fractional complex Ginzburg-Landau equation,, Int. J. Bifur. Chaos, 23 (2013). doi: 10.1142/S0218127413502027. Google Scholar

[11]

H. Lu and S. Lü, Random attractor for fractional Ginzburg-Landau equation with multiplicative noise,, Taiwanese J. Math., 18 (2014), 435. doi: 10.11650/tjm.18.2014.3053. Google Scholar

[12]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach,, Phys. Rep., 339 (): 1. doi: 10.1016/S0370-1573(00)00070-3. Google Scholar

[13]

E. W. Montroll and M. F. Shlesinger, On the wonderful world of random walks,, Nonequilibrium phenomena, (1984), 1. Google Scholar

[14]

R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry,, Phys. Stat. Solidi. B, 133 (1986), 425. doi: 10.1002/pssb.2221330150. Google Scholar

[15]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115. Google Scholar

[16]

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

[17]

X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation,, Appl. Anal., 92 (2013), 318. doi: 10.1080/00036811.2011.614601. Google Scholar

[18]

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

[19]

M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange kinetics,, Nature, 363 (1993), 31. doi: 10.1038/363031a0. Google Scholar

[20]

Y. Su and Z. Feng, Existence theory for an arbitrary order fractional differential equation with deviating argument,, Acta Appl. Math., 118 (2012), 81. doi: 10.1007/s10440-012-9679-1. Google Scholar

[21]

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

[22]

R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[23]

S. Wheatcraft and M. Meerschaert, Fractional conservation of mass,, Advances in Water Resources, 31 (2008), 1377. doi: 10.1016/j.advwatres.2008.07.004. Google Scholar

[24]

G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics,, Reprint of the 2005 original. Oxford University Press, (2005). Google Scholar

[25]

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

show all references

References:
[1]

H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynamics and Differential Equations, 9 (1997), 307. doi: 10.1007/BF02219225. Google Scholar

[2]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar

[3]

A. Debussche, Hausdorff dimension of a random invariant set,, J. Math. Pure Appl., 77 (1998), 967. doi: 10.1016/S0021-7824(99)80001-4. Google Scholar

[4]

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

[5]

C. W. Gardiner, Handbooks of Stochastic Methods for Physics,, Chemistry and Natural Sciences, (1983). doi: 10.1007/978-3-662-02377-8. Google Scholar

[6]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation,, Commun. Partial Differential Equations, 36 (2011), 247. doi: 10.1080/03605302.2010.503769. Google Scholar

[7]

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

[8]

S. Holm and S. P. Näsholm, Comparison of fractional wave equations for power law attenuation in ultrasound and elastography,, Ultrasound Med. Biol., 40 (2014), 695. doi: 10.1016/j.ultrasmedbio.2013.09.033. Google Scholar

[9]

N. Laskin, Fractional Schrödinger equation,, Physical Review E, 66 (2002). doi: 10.1103/PhysRevE.66.056108. Google Scholar

[10]

H. Lu, S. Lü and Z. Feng, Asymptotic dynamics of 2d fractional complex Ginzburg-Landau equation,, Int. J. Bifur. Chaos, 23 (2013). doi: 10.1142/S0218127413502027. Google Scholar

[11]

H. Lu and S. Lü, Random attractor for fractional Ginzburg-Landau equation with multiplicative noise,, Taiwanese J. Math., 18 (2014), 435. doi: 10.11650/tjm.18.2014.3053. Google Scholar

[12]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach,, Phys. Rep., 339 (): 1. doi: 10.1016/S0370-1573(00)00070-3. Google Scholar

[13]

E. W. Montroll and M. F. Shlesinger, On the wonderful world of random walks,, Nonequilibrium phenomena, (1984), 1. Google Scholar

[14]

R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry,, Phys. Stat. Solidi. B, 133 (1986), 425. doi: 10.1002/pssb.2221330150. Google Scholar

[15]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115. Google Scholar

[16]

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

[17]

X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation,, Appl. Anal., 92 (2013), 318. doi: 10.1080/00036811.2011.614601. Google Scholar

[18]

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

[19]

M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange kinetics,, Nature, 363 (1993), 31. doi: 10.1038/363031a0. Google Scholar

[20]

Y. Su and Z. Feng, Existence theory for an arbitrary order fractional differential equation with deviating argument,, Acta Appl. Math., 118 (2012), 81. doi: 10.1007/s10440-012-9679-1. Google Scholar

[21]

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

[22]

R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[23]

S. Wheatcraft and M. Meerschaert, Fractional conservation of mass,, Advances in Water Resources, 31 (2008), 1377. doi: 10.1016/j.advwatres.2008.07.004. Google Scholar

[24]

G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics,, Reprint of the 2005 original. Oxford University Press, (2005). Google Scholar

[25]

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

[1]

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

[2]

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

[3]

N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647

[4]

Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173

[5]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871

[6]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665

[7]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359

[8]

Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311

[9]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian, Meisheng Li. Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 507-527. doi: 10.3934/dcdsb.2011.16.507

[15]

Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579

[16]

Michael Stich, Carsten Beta. Standing waves in a complex Ginzburg-Landau equation with time-delay feedback. Conference Publications, 2011, 2011 (Special) : 1329-1334. doi: 10.3934/proc.2011.2011.1329

[17]

Boling Guo, Bixiang Wang. Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 455-466. doi: 10.3934/dcds.1996.2.455

[18]

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

[19]

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

[20]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]