A topological approximation of the nonlinear Anderson model

Abstract

We study the phenomena of Anderson localization in the presence of nonlinear interaction on a lattice. A class of nonlinear Schrodinger models with arbitrary power nonlinearity is analyzed. We conceive the various regimes of behavior, depending on the topology of resonance-overlap in phase space, ranging from a fully developed chaos involving Levy flights to pseudochaotic dynamics at the onset of delocalization. It is demonstrated 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. We describe this localization-delocalization transition as a percolation transition on a Cayley tree. It is found in vicinity of the criticality that the spreading of the wave field is subdiffusive in the limit t\rightarrow+\infty. The second moment grows with time as a powerlaw t^α, with α= 1/3. Also we find for superquadratic nonlinearity that the analog pseudochaotic regime at the edge of chaos is self-controlling in that it has feedback on the topology of the structure on which the transport processes concentrate. Then the system automatically (without tuning of parameters) develops its percolation point. We classify this type of behavior in terms of self-organized criticality dynamics in Hilbert space. For subquadratic nonlinearities, the behavior is shown to be sensitive to details of definition of the nonlinear term. A transport model is proposed based on modified nonlinearity, using the idea of stripes propagating the wave process to large distances. Theoretical investigations, presented here, are the basis for consistency analysis of the different localization-delocalization patterns in systems with many coupled degrees of freedom in association with the asymptotic properties of the transport.

20 pages, 2 figures; improved text with revisions; accepted for publication in Physical Review E

References in corpus (9)

• Direct observation of Anderson localization of matter-waves in a controlled disorder
• Destruction of Anderson localization by a weak nonlinearity
• Universal spreading of wavepackets in disordered nonlinear systems
• Rigorous Derivation of the Gross-Pitaevskii Equation
• Fractional generalization of the Ginzburg-Landau equation: An unconventional approach to critical phenomena in complex media
• Expansion of a Bose-Einstein Condensate in the Presence of Disorder
• Fractional Fokker-Planck equation for Lévy flights in nonhomogeneous environments
• Pseudochaos and low-frequency percolation scaling for turbulent diffusion in magnetized plasma
• A mixed SOC-turbulence model for nonlocal transport and Levy-fractional Fokker-Planck equation

Cited by in corpus (5)

• 25 Years of Self-Organized Criticality: Solar and Astrophysics
• Nonlinear lattice waves in heterogeneous media
• Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method
• Subdiffusive Lévy flights in quantum nonlinear Schrödinger lattices with algebraic power nonlinearity
• Subdiffusion in classical and quantum nonlinear Schrödinger equations with disorder