Discontinuity, Nonlinearity, and Complexity
Topology of Delocalization in the Nonlinear Anderson Model and Anomalous Diffusion on Finite Clusters
Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 151162  DOI:10.5890/DNC.2015.06.003
A.V. Milovanov$^{1}$,$^{2}$,$^{4}$; A. Iomin$^{3}$,$^{4}$
$^{1}$ ENEA National Laboratory, Centro Ricerche Frascati, I00044 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
$^{4}$ MaxPlanckInstitut 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 powerlaw 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 localizationdelocalization transition with unlimited spreading already at the delocalization border. For superquadratic 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 halfinteger, 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 nonMarkovian dynamics with algebraically decaying time correlations.
Acknowledgments
A.V.M. and A.I. thank theMaxPlanckInstitute 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 “SelfOrganized Criticality and Turbulence” (Bern, Switzerland).
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