Free Access
Issue
Math. Model. Nat. Phenom.
Volume 8, Number 2, 2013
Anomalous diffusion
Page(s) 88 - 99
DOI https://doi.org/10.1051/mmnp/20138206
Published online 24 April 2013
  1. P.W. Anderson. Absence of diffusion in certain random lattices. Phys. Rev. 109 (1958), 1492-1505. [NASA ADS] [CrossRef] [Google Scholar]
  2. Yu.Yu. Bagderina. Equivalence of ordinary differential equations. Differential Equations, 43 (2007), 595-604. [CrossRef] [MathSciNet] [Google Scholar]
  3. R. Bekenstein, M. Segev, Self-accelerating optical beams in highly nonlocal nonlinear media. Optics Express, 19 (2011), No. 24, 23706-23715. [CrossRef] [PubMed] [Google Scholar]
  4. D. ben-Avraam, S. Havlin. Diffusion and Reactions in Fractals and Disordered Systems. University Press, Cambridge, 2000. [Google Scholar]
  5. M.V. Berry, N.L. Balazs. Nonspreading wave packets. Am. J. Phys., 47 (1979) No. 3, 264-267. [CrossRef] [Google Scholar]
  6. J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clťement, L. Sanchez-Palencia, P. Bouyer, A. Aspect. Direct observation of Anderson localization of matter waves in a controlled disorder. Nature, 453, (2008), 891-894. [CrossRef] [PubMed] [Google Scholar]
  7. B.V. Chirikov, V.V. Vecheslavov. Arnold diffusion in large systems. Sov. Phys. JETP, 85 (1997), 616 [Zh. Eksp. Teor. Fiz. 112 (1997), 1132]. [CrossRef] [Google Scholar]
  8. H.T. Davis. Introduction to Nonlinear Differential and Integral Equations. Dover Publications Inc., New York, 1962. [Google Scholar]
  9. D. Emin, C.F. Hart. Existence of Wannier-Stark localization. Phys. Rev. B, 36 (1987), 7353-7359. [CrossRef] [Google Scholar]
  10. S. Flach. Spreading of waves in nonlinear disordered media. Chem. Phys., 375 (2010), 548-556. [CrossRef] [Google Scholar]
  11. S. Flach, D.O. Krimer, Ch. Skokos. Universal spreading of wave packets in disordered nonlinear systems. Phys. Rev. Lett., 102 (2008), 024101. [CrossRef] [PubMed] [Google Scholar]
  12. H. Fukuyama, R.A. Bari, H.C. Fogedby. Tightly bond electrons in a uniform electric field. Phys. Rev. B, 8 (1973), 5579-5586. [CrossRef] [Google Scholar]
  13. Y. He, S. Burov, R. Metzler, E. Barkai. Random Time-Scale Invariant Diffusion and Transport Coefficients. Phys. Rev. Lett., 101 (2008), 058101. [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
  14. E.L. Ince. Ordinary Differential Equations. Longmans, Green and CO. LTD., London, 1927. [Google Scholar]
  15. A. Iomin. Subdiffusion in the nonlinear Schrödinger equation with disorder. Phys. Rev. E, 81 (2010), 017601. [CrossRef] [Google Scholar]
  16. A. Iomin. Dynamics of wave packets for the nonlinear Schrödinger equation with a random potential. Phys. Rev. E, 80 (2009), 037601. [CrossRef] [Google Scholar]
  17. E. Janke, F. Emde, F. Lösch. Tafeln Höherer Functionen. B.G. Taubner Verlagsgesellschaft, Stuttgart, 1960. [Google Scholar]
  18. I. Kramer, M. Segev, D.N. Christodoulides. Self-accelerating self-trapped optical beams. Phys. Rev. Lett., 106 (20110), 213903. [Google Scholar]
  19. I. Kramer, Y. Lamer, M. Segev, D.N. Christodoulides. Causality effects on accelerating light pulses. Optics Express, 19 (23) (2011), 23132–23139. [CrossRef] [PubMed] [Google Scholar]
  20. A.R. Kolovsky, E.A. Gómez, H.J. Korsh. Bose-Einstein condensates on tilted lattices: Coherent, chaotic, and subdiffusive dynamics. Phys. Rev. A, 81 (2010), 025603. [CrossRef] [Google Scholar]
  21. D.O. Krimer, R. Khomeriki, S. Flach. Delocalization and spreading in a nonlinear Stark ladder. Phys. Rev. E, 80 (2009), 036201. [CrossRef] [Google Scholar]
  22. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D.N. Christodoulides, Y. Silberberg. Anderson localization and nonlinearity in one-dimensional disordered photonic lattices. Phys. Rev. Lett., 100 (2008), 013906. [CrossRef] [PubMed] [Google Scholar]
  23. I.M. Lifshits, S.A. Gredeskul, and L.A. Pastur. Introduction to the Theory of Disordered Systems. Wiley-Interscience, New York, 1988. [Google Scholar]
  24. O. Lyubomudrov, M. Edelman, G.M. Zaslavsky. Pseudochaotic systems and their fractional kinetics. Intl. J. Modern Phys. B 17 (2003), 4149-4167. [CrossRef] [Google Scholar]
  25. F. Mainardi. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals, 7(9) (1996), 1461–1477. [CrossRef] [MathSciNet] [Google Scholar]
  26. R. Metzler. Generalized Chapman-Kolmogorov equation: A unifying approach to the description of anomalous transport in external fields. Phys. Rev. E, 62 (2000), 6233-6245. [CrossRef] [MathSciNet] [Google Scholar]
  27. R. Metzler, J. Klafter. The Random Walk’s Guide to Anomalous Diffusion: a Fractional Dynamics Approach. Phys. Rep., 339 (2000), 1-77. [NASA ADS] [CrossRef] [Google Scholar]
  28. A.V. Milovanov. Pseudochaos and low-frequency percolation scaling for turbulent diffusion in magnetized plasma. Phys. Rev. E, 79 (2009), 046403. [NASA ADS] [CrossRef] [Google Scholar]
  29. A.V. Milovanov, A. Iomin. Localization-delocalization transition on a separatrix system of nonlinear Schrödinger equation with disorder. Europhys. Lett., 100 (2012), 10006. [CrossRef] [Google Scholar]
  30. M.I. Molina. Transport of localized and extended excitations in a nonlinear Anderson model. Phys. Rev. B, 58 (1998), 12547. [CrossRef] [Google Scholar]
  31. E.W. Montroll and M.F. Shlesinger. The wonderful wold of random walks. In Studies in Statistical Mechanics , v. 11, eds J. Lebowitz and E.W. Montroll. North–Holland, Amsterdam, 1984. [Google Scholar]
  32. E.W. Montroll, G.H. Weiss, Random walks on lattices. II J. Math. Phys., 6 (1965), 167; [Google Scholar]
  33. E.W. Montroll. Random walks on lattices III. Calculation of first passage times with application to exciton trapping on photosynthetic units. J. Math. Phys., 10 (1969), 753. [CrossRef] [Google Scholar]
  34. M. Mulansky, K. Ahnert, A. Pikovsky, D.L. Shepelyansky. Strong and weak chaos in weakly nonintegrable many-body Hamiltonian systems. J. Stat. Phys., 145 (2011), 1256-1274. [CrossRef] [MathSciNet] [Google Scholar]
  35. K.B. Oldham and J. Spanier. The Fractional Calculus. Academic Press, Orlando, 1974. [Google Scholar]
  36. F.W.J. Olver. Asymptotics and Special Function. Academic Press, New York, 1974. [Google Scholar]
  37. A.S. Pikovsky and D.L. Shepelyansky. Destruction of Anderson localization by a weak nonlinearity. Phys. Rev. Lett., 100 (2008), 094101. [CrossRef] [PubMed] [Google Scholar]
  38. O.J. Pine, D.A. Weitz, P.M. Chaikin, E. Herbolzheimer. Diffusing wave spectroscopy. Phys Rev Lett., 60 (1988), 1134-1137. [CrossRef] [PubMed] [Google Scholar]
  39. I. Podlubny. Fractional Differential Equations. Academic Press, San Diego, 1999. [Google Scholar]
  40. T. Schwartz, G. Bartal, S. Fishman, M. Segev. Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature, 446 (2007), 52. [CrossRef] [PubMed] [Google Scholar]
  41. D.L. Shepelyansky. Delocalization of quantum chaos by weak nonlinearity. Phys. Rev. Lett., 70 (1993), 1787. [CrossRef] [PubMed] [Google Scholar]
  42. N.N. Tarkhanov, private communication. [Google Scholar]
  43. G.H. Wannier. Wave functions and effective Hamiltonian for Bloch electrons in an electric field. Phys. Rev., 117 (1960), 432-439. [CrossRef] [MathSciNet] [Google Scholar]
  44. G.M. Zaslavsky. Statistical Irreversibility in Non- linear Systems. Nauka, Moscow, 1970. [Google Scholar]
  45. G.M. Zaslavsky. Fractional kinetic equation for Hamiltonian chaos. Physica D, 76, (1994), 110-122. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.