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 and 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), 307-341. doi: 10.1007/BF02219225.

[2]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

[3]

A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pure Appl., 77 (1998), 967-988. doi: 10.1016/S0021-7824(99)80001-4.

[4]

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.

[5]

C. W. Gardiner, Handbooks of Stochastic Methods for Physics, Chemistry and Natural Sciences, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-662-02377-8.

[6]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differential Equations, 36 (2011), 247-255. doi: 10.1080/03605302.2010.503769.

[7]

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.

[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), 695-703. doi: 10.1016/j.ultrasmedbio.2013.09.033.

[9]

N. Laskin, Fractional Schrödinger equation, Physical Review E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.

[10]

H. Lu, S. Lü and Z. Feng, Asymptotic dynamics of 2d fractional complex Ginzburg-Landau equation, Int. J. Bifur. Chaos, 23 (2013), 1350202, 12pp. doi: 10.1142/S0218127413502027.

[11]

H. Lu and S. Lü, Random attractor for fractional Ginzburg-Landau equation with multiplicative noise, Taiwanese J. Math., 18 (2014), 435-450. doi: 10.11650/tjm.18.2014.3053.

[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.

[13]

E. W. Montroll and M. F. Shlesinger, On the wonderful world of random walks, Nonequilibrium phenomena, II, North-Holland, Amsterdam, 1984, 1-121.

[14]

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.

[15]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.

[16]

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.

[17]

X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Appl. Anal., 92 (2013), 318-334. doi: 10.1080/00036811.2011.614601.

[18]

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

[19]

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

[20]

Y. Su and Z. Feng, Existence theory for an arbitrary order fractional differential equation with deviating argument, Acta Appl. Math., 118 (2012), 81-105. doi: 10.1007/s10440-012-9679-1.

[21]

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.

[22]

R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[23]

S. Wheatcraft and M. Meerschaert, Fractional conservation of mass, Advances in Water Resources, 31 (2008), 1377-1381. doi: 10.1016/j.advwatres.2008.07.004.

[24]

G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Reprint of the 2005 original. Oxford University Press, Oxford, 2008.

[25]

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]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynamics and Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.

[2]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

[3]

A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pure Appl., 77 (1998), 967-988. doi: 10.1016/S0021-7824(99)80001-4.

[4]

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.

[5]

C. W. Gardiner, Handbooks of Stochastic Methods for Physics, Chemistry and Natural Sciences, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-662-02377-8.

[6]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differential Equations, 36 (2011), 247-255. doi: 10.1080/03605302.2010.503769.

[7]

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.

[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), 695-703. doi: 10.1016/j.ultrasmedbio.2013.09.033.

[9]

N. Laskin, Fractional Schrödinger equation, Physical Review E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.

[10]

H. Lu, S. Lü and Z. Feng, Asymptotic dynamics of 2d fractional complex Ginzburg-Landau equation, Int. J. Bifur. Chaos, 23 (2013), 1350202, 12pp. doi: 10.1142/S0218127413502027.

[11]

H. Lu and S. Lü, Random attractor for fractional Ginzburg-Landau equation with multiplicative noise, Taiwanese J. Math., 18 (2014), 435-450. doi: 10.11650/tjm.18.2014.3053.

[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.

[13]

E. W. Montroll and M. F. Shlesinger, On the wonderful world of random walks, Nonequilibrium phenomena, II, North-Holland, Amsterdam, 1984, 1-121.

[14]

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.

[15]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.

[16]

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.

[17]

X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Appl. Anal., 92 (2013), 318-334. doi: 10.1080/00036811.2011.614601.

[18]

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

[19]

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

[20]

Y. Su and Z. Feng, Existence theory for an arbitrary order fractional differential equation with deviating argument, Acta Appl. Math., 118 (2012), 81-105. doi: 10.1007/s10440-012-9679-1.

[21]

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.

[22]

R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[23]

S. Wheatcraft and M. Meerschaert, Fractional conservation of mass, Advances in Water Resources, 31 (2008), 1377-1381. doi: 10.1016/j.advwatres.2008.07.004.

[24]

G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Reprint of the 2005 original. Oxford University Press, Oxford, 2008.

[25]

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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