Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T14:35:23.216Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics in Nonlinear Schrödinger Equation with dc bias: From Subdiffusion to Painlevé Transcendent

Published online by Cambridge University Press:  24 April 2013

A. Iomin
Get access

Abstract

Dynamics of the nonlinear Schrödinger equation in the presence of a constant electric field is studied. Both discrete and continuous limits of the model are considered. For the discrete limit, a probabilistic description of subdiffusion is suggested and a subdiffusive spreading of a wave packet is explained in the framework of a continuous time random walk. In the continuous limit, the biased nonlinear Schrödinger equation is shown to be integrable, and solutions in the form of the Painlevé transcendents are obtained.

Keywords

continuous time random walksubdiffusionfractional Fokker-Planck equationStark ladderself-accelerating solution
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 2: Anomalous diffusion , 2013 , pp. 88 - 99
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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