The European Physical Journal B

, Volume 84, Issue 1, pp 79–82 | Cite as

Influence of weak nonlinearity on the 1D Anderson model with long-range correlated disorder

Regular Article Mesoscopic and Nanoscale Systems


We investigate theoretically the nature of the states and the localization properties in a one-dimensional Anderson model with long-range correlated disorder and weak nonlinearity. Using the stationary discrete nonlinear Schrödinger equation, we calculate the disorder-averaged logarithm of the transmittance and the localization length in the fixed input case in a numerically exact manner. Unlike in many previous studies, we strictly fix the intensity of the incident wave and calculate the localization length as a function of other parameters. We also calculate the wave functions in a given disorder configuration. In the linear case, flat phased localized states appear near the bottom of the band and staggered localized states appear near the top of the band, while a continuum of extended states appears near the band center. We find that the focusing Kerr-type nonlinearity enhances the Anderson localization of flat phased states and suppresses that of staggered states. We observe that there exists a perfect symmetry relationship for the localization length between focusing and defocusing nonlinearities. Above a critical value of the strength of nonlinearity, delocalization due to the long-range correlations of disorder is destroyed and all states become localized.


Localization Length Band Center Anderson Localization Weak Nonlinearity Input Case 
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  1. 1.
    P.A. Lee, T.V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985)ADSCrossRefGoogle Scholar
  2. 2.
    I.M. Lifshits, S.A. Gredeskul, L.A. Pastur, Introduction to the Theory of Disordered Systems (New York, Wiley, 1988)Google Scholar
  3. 3.
    D.H. Dunlap, H.-L. Wu, P.W. Phillips, Phys. Rev. Lett. 65, 88 (1990)ADSCrossRefGoogle Scholar
  4. 4.
    F.A.B.F. de Moura, M.L. Lyra, Phys. Rev. Lett. 81, 3735 (1998)ADSCrossRefGoogle Scholar
  5. 5.
    F.M. Izrailev, A.A. Krokhin, Phys. Rev. Lett. 82, 4062 (1999)ADSCrossRefGoogle Scholar
  6. 6.
    F. Domínguez-Adame, V.A. Malyshev, F.A.B.F. de Moura, M.L. Lyra, Phys. Rev. Lett. 91, 197402 (2003)ADSCrossRefGoogle Scholar
  7. 7.
    H. Shima, T. Nomura, T. Nakayama, Phys. Rev. B 70, 075116 (2004)ADSCrossRefGoogle Scholar
  8. 8.
    T. Kaya, Eur. Phys. J. B 55, 49 (2007)ADSCrossRefGoogle Scholar
  9. 9.
    S. Nishino, K. Yakubo, H. Shima, Phys. Rev. B 79, 033105 (2009)ADSCrossRefGoogle Scholar
  10. 10.
    Y. Zhao, S. Duan, W. Zhang, Physica E 42, 1425 (2010)ADSCrossRefGoogle Scholar
  11. 11.
    V. Bellani, E. Diez, R. Hey, L. Toni, L. Tarricone, G.B. Parravicini, F. Domínguez-Adame, R. Gómez-Alcalá, Phys. Rev. Lett. 82, 2159 (1999)ADSCrossRefGoogle Scholar
  12. 12.
    U. Kuhl, F.M. Izrailev, A.A. Krokhin, H.-J. Stöcmann, Appl. Phys. Lett. 77, 633 (2000)ADSCrossRefGoogle Scholar
  13. 13.
    B.P. Nguyen, K. Kim, F. Rotermund, H. Lim, Physica B 406, 4535 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    B. Doucot, R. Rammal, Europhys. Lett. 3, 969 (1987)ADSCrossRefGoogle Scholar
  15. 15.
    S.A. Gredeskul, Y.S. Kivshar, Phys. Rep. 216, 1 (1992)MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    D. Hennig, G.P. Tsironis, Phys. Rep. 307, 333 (1999)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    P.G. Kevrekidis, K.Ø. Rasmussen, A.R. Bishop, Int. J. Mod. Phys. B 15, 2833 (2001)ADSCrossRefGoogle Scholar
  18. 18.
    T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, F. Lederer, Phys. Rev. Lett. 93, 053901 (2004)ADSCrossRefGoogle Scholar
  19. 19.
    Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D.N. Christodoulides, Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    A.S. Pikovsky, D.L. Shepelyansky, Phys. Rev. Lett. 100, 094101 (2008)ADSCrossRefGoogle Scholar
  21. 21.
    F. Delyon, Y.-E. Lévy, B. Souillard, Phys. Rev. Lett. 57, 2010 (1986)ADSCrossRefGoogle Scholar
  22. 22.
    Y. Wan, C.M. Soukoulis, Phys. Rev. A 41, 800 (1990)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    M.I. Molina, G.P. Tsironis, Phys. Rev. Lett. 73, 464 (1994)ADSCrossRefGoogle Scholar
  24. 24.
    A. Tietsche, A. Pikovsky, Europhys. Lett. 84, 10006 (2008)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Division of Energy Systems ResearchAjou UniversitySuwonKorea

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