The nonlinear Schrödinger equation with a random potential: results and puzzles

Fishman, Shmuel; Krivolapov, Yevgeny; Soffer, Avy

doi:10.1088/0951-7715/25/4/r53

Cited by 88 publications

(101 citation statements)

References 70 publications

(186 reference statements)

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“…One of the hotly debated questions in the community is whether the subdiffusive spreading will continue forever or eventually slow down, or even stop (see e.g. [72] and references therein). This is an interesting and perhaps mathematically deep question, despite the absence of rigorous results which would fuel the above doubts.…”

Section: Discussionmentioning

confidence: 99%

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Nonlinear Lattice Waves in Random Potentials

Flach

2015

Nonlinear Optical and Atomic Systems

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Localization of waves by disorder is a fundamental physical problem encompassing a diverse spectrum of theoretical, experimental and numerical studies in the context of metal-insulator transition, quantum Hall effect, light propagation in photonic crystals, and dynamics of ultra-cold atoms in optical arrays. Large intensity light can induce nonlinear response, ultracold atomic gases can be tuned into an interacting regime, which leads again to nonlinear wave equations on a mean field level. The interplay between disorder and nonlinearity, their localizing and delocalizing effects is currently an intriguing and challenging issue in the field. We will discuss recent advances in the dynamics of nonlinear lattice waves in random potentials. In the absence of nonlinear terms in the wave equations, Anderson localization is leading to a halt of wave packet spreading. Nonlinearity couples localized eigenstates and, potentially, enables spreading and destruction of Anderson localization due to nonintegrability, chaos and decoherence. The spreading process is characterized by universal subdiffusive laws due to nonlinear diffusion. We review extensive computational studies for one-and two-dimensional systems with tunable nonlinearity power. We also briefly discuss extensions to other cases where the linear wave equation features localization: Aubry-Andre localization with quasiperiodic potentials, Wannier-Stark localization with dc fields, and dynamical localization in momentum space with kicked rotors.

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Section: Discussionmentioning

confidence: 99%

“…in [16], [74] have connectivity K = 2. Again the strong disorder limit of K = 4 models yields K = 2 in leading order, which is one of the cases where analytical methods are applied (see references in [72]). Time might be ripe to perform comparative studies.…”

Section: Discussionmentioning

confidence: 99%

Nonlinear Lattice Waves in Random Potentials

Flach

2015

Nonlinear Optical and Atomic Systems

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“…6 The mechanisms and details of these processes remain inconclusive and the contradictions are hotly debated. 8 Understanding the competition between disorder and interactions promises a wealth of applications because in any realistic system both are always present.…”

Section: Introductionmentioning

confidence: 99%

Flat band states: Disorder and nonlinearity

et al. 2013

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We study the critical behavior of Anderson localized modes near intersecting flat and dispersive bands in the quasi-one-dimensional diamond ladder with weak diagonal disorder W . The localization length ξ of the flat band states scales with disorder as ξ ∼ W −γ , with γ ≈ 1.3, in contrast to the dispersive bands with γ = 2. A small fraction of dispersive modes mixed with the flat band states is responsible for the unusual scaling. Anderson localization is therefore controlled by two different length scales. Nonlinearity can produce qualitatively different wave spreading regimes, from enhanced expansion to resonant tunneling and self-trapping.

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“…The early theoretical and numerical studies [2,3] have been followed by more recent and more detailed analysis performed by different groups [4,5,6,7,8,9], [10,11,12,13,14,15]. The interest to this problem comes also from the side of mathematics which puts forward a fundamental question on how the pure point spectrum of Anderson localization is affected by a weak nonlinearity [16,17,18]. At the same time the experiments on spreading of light in nonlinear photonic lattices [19,20] and of Bose-Einstein condensates of cold atoms in disordered potential [21] start to be able to observe effects of nonlinearity on localization.…”

Section: Introductionmentioning

confidence: 99%

Destruction of Anderson localization by nonlinearity in kicked rotator at different effective dimensions

Ermann

Shepelyansky

2014

J. Phys. A: Math. Theor.

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Abstract. We study numerically the frequency modulated kicked nonlinear rotator with effective dimension d = 1, 2, 3, 4. We follow the time evolution of the model up to 10 9 kicks and determine the exponent α of subdiffusive spreading which changes from 0.35 to 0.5 when the dimension changes from d = 1 to 4. All results are obtained in a regime of relatively strong Anderson localization well below the Anderson transition point existing for d = 3, 4. We explain that this variation of the exponent is different from the usual d−dimensional Anderson models with local nonlinearity where α drops with increasing d. We also argue that the renormalization arguments proposed by Cherroret N et al. arXiv:1401.1038 are not valid.

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The nonlinear Schrödinger equation with a random potential: results and puzzles

Cited by 88 publications

References 70 publications

Nonlinear Lattice Waves in Random Potentials

Nonlinear Lattice Waves in Random Potentials

Flat band states: Disorder and nonlinearity

Destruction of Anderson localization by nonlinearity in kicked rotator at different effective dimensions

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