Localization-delocalization transition on a separatrix system of nonlinear Schrödinger equation with disorder

Milovanov, Alexander V.; Iomin, Alexander

doi:10.1209/0295-5075/100/10006

Cited by 31 publications

(87 citation statements)

References 32 publications

(81 reference statements)

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“…Also we predict that the transport of waves at the border of delocalization is subdiffusive, with the exponent α which is inversely proportional with the power nonlinearity increased by one. For quadratic nonlinearity we have (∆n) 2 (t) ∝ t 1/3 for t → +∞ consistently with the previous investigations [10,11]. 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.…”

Section: Discussionsupporting

confidence: 83%

“…We reiterate that non-diagonal elements V k,m1,m2,m3 characterize couplings between each four eigenstates with wave numbers k, m 1 , m 2 , and m 3 . The comprehension of Hamiltonian character of the dynamics paves the way for a consistency analysis of the various transport scenarios behind the Anderson localization problem (with the topology of resonance overlap taken into account) [10,11]. To this end, the transport problem for the wave function becomes essentially a topological problem in phase space.…”

Section: The Three-step Topological Approachmentioning

confidence: 99%

“…It has been discussed by a few authors [8][9][10] that NLSE with quadratic nonlinearity (i.e., s = 1) observes a localization-delocalization transition above a certain critical strength of nonlinear interaction. That means that the localized state is destroyed, and the nonlinear field can spread across the lattice despite the underlying disorder, provided just that the β value exceeds a maximal allowed value.…”

Section: Introductionmentioning

confidence: 99%

“…This localization-delocalization transition bears signatures, enabling to associate it with a percolation transition on the infinite Cayley tree (Bethe lattice) [10,11]. The main idea here is that delocalization occurs through infinite clusters of chaotic states on a Bethe lattice, with occupancy probabilities decided by the strength of nonlinear interaction.…”

Section: Introductionmentioning

confidence: 99%

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

Milovanov¹,

Iomin²

2015

DNC

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This study is concerned with destruction of Anderson localization by a nonlinearity of the powerlaw type. We suggest using a nonlinear Schrödinger 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 nonlinear Anderson 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 non-Markovian dynamics with algebraically decaying time correlations.

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Section: Discussionsupporting

confidence: 83%

Section: The Three-step Topological Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Milovanov¹,

Iomin²

2015

DNC

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“…We analyze delocalization processes as a transport problem for a dynamical system with many coupled degrees of freedom, with an emphasis on the criticality aspects of delocalization. A short account of this approach has been reported previously [20].…”

Section: Introductionmentioning

confidence: 99%

Topological approximation of the nonlinear Anderson model

Milovanov

Iomin

2014

Phys. Rev. E

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We study the phenomena of Anderson localization in the presence of nonlinear interaction on a lattice. A class of nonlinear Schrödinger 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 Lévy flights to pseudochaotic dynamics at the onset of delocalization. It is demonstrated that the 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. We describe this localization-delocalization transition as a percolation transition on the infinite Cayley tree (Bethe lattice). It is found in vicinity of the criticality that the spreading of the wave field is subdiffusive in the limit t → +∞. The second moment of the associated probability distribution grows with time as a powerlaw ∝ t α , with the exponent α = 1/3 exactly. 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 (SOC) 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.

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25 Years of Self-Organized Criticality: Solar and Astrophysics

et al. 2014

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Shortly after the seminal paper "Self-Organized Criticality: An explanation of 1/f noise" by Bak et al. (1987), the idea has been applied to solar physics, in "Avalanches and the Distribution of Solar Flares" by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme

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Localization-delocalization transition on a separatrix system of nonlinear Schrödinger equation with disorder

Cited by 31 publications

References 32 publications

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

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

Topological approximation of the nonlinear Anderson model

25 Years of Self-Organized Criticality: Solar and Astrophysics

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