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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Topology of Delocalization in the Nonlinear Anderson Model and Anomalous Diffusion on Finite Clusters

Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 151--162 | DOI:10.5890/DNC.2015.06.003

A.V. Milovanov$^{1}$,$^{2}$,$^{4}$; A. Iomin$^{3}$,$^{4}$

$^{1}$ ENEA National Laboratory, Centro Ricerche Frascati, I-00044 Frascati, Rome, Italy

$^{2}$ Space Research Institute, Russian Academy of Sciences, 117997 Moscow, Russia

$^{3}$ Department of Physics and Solid State Institute, Technion, Haifa 32000, Israel

$^{4}$ Max-Planck-Institut f¨ur Physik komplexer Systeme, 01187 Dresden, Germany

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Abstract

This study is concernedwith destruction of Anderson localization by a nonlinearity of the power-law type. We suggest using a nonlinear Schr¨odinger model with random potential on a lattice that quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. For super-quadratic nonlinearity the borderline spreading corresponds to diffusion processes on finite clusters. We have proposed an analytical method to predict and explain such transport processes. Our method uses a topological approximation of the nonlinearAnderson model and, if the exponent of the power nonlinearity is either integer or half-integer, will yield the wanted value of the transport exponent via a triangulation procedure in an Euclidean mapping space. A kinetic picture of the transport arising from these investigations uses a fractional extension of the diffusion equation to fractional derivatives over the time, signifying non-Markovian dynamics with algebraically decaying time correlations.

Acknowledgments

A.V.M. and A.I. thank theMax-Planck-Institute for the Physics of Complex Systems for hospitality and financial support. This work was supported in part by the Israel Science Foundation (ISF) and by the ISSI project “Self-Organized Criticality and Turbulence” (Bern, Switzerland).

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