Barrier transmission for the nonlinear Schrödinger equation: surprises of nonlinear transport

Rapedius, Kevin; Korsch, Hans Jürgen

doi:10.1088/1751-8113/41/35/355001

Cited by 4 publications

(6 citation statements)

References 11 publications

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“…away from the resonance) on the order of ν typ ∼ 5 × 10 −8 -at the resonance it is however ν res = 1.5 × 10 −5 . These findings are similar to previous work on nonlinear resonant scattering from quantum dots [3,4] and from one-dimensional structures [5][6][7]. Our model generalises the latter results by allowing for additional topological complexity.…”

Section: Implications For Nonlinear Scattering: Multistabilitysupporting

confidence: 90%

“…In any generic model of chaotic scattering unimodular eigenvalues of the subunitary matrix σ int (which acts on a vector of 2B coefficients, one for each directed bond) are strongly suppressed. For other known examples of resonant scattering in one-dimensional nonlinear Schrödinger systems [5][6][7], the equivalent of σ int is one number with modulus smaller than 1. In all these models high amplification factors are either cut-off or extremely rare.…”

Section: B Linear Scattering: Resonances and Amplificationmentioning

confidence: 99%

“…The theory will be applied in particular to show the effect of nonlinearity on transmission through networks of nonlinear fibers. Our model may also be used as a simple yet non-trivial model where the universal properties derived from detailed numerical computations of Bose-Einstein condensates in non-regular traps [2][3][4][5][6][7] could be further investigated.Scattering is studied as a stationary process. The main finding is that the sharp resonances which dominate scattering in networks with complex connectivity lead to a dramatic amplification of the nonlinear effects: while non-resonant scattering hardly deviates from the predictions of the linear theory, tuning the parameters to a nearby resonance (without changing the incoming field intensity) brings the system into the nonlinear regime which is signaled by multi-stability and hysteresis.…”

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confidence: 99%

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confidence: 99%

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Stationary scattering from a nonlinear network

2011

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Transmission through a complex network of nonlinear one-dimensional leads is discussed by extending the stationary scattering theory on quantum graphs to the nonlinear regime. We show that the existence of cycles inside the graph leads to a large number of sharp resonances that dominate scattering. The latter resonances are then shown to be extremely sensitive to the nonlinearity and display multi-stability and hysteresis. This work provides a framework for the study of light propagation in complex optical networks.The study of quantum graphs has gained its popularity in recent years [1] not only because graphs emulate successfully complex mesoscopic and optical networks, but also because they manage to reproduce universal properties (such as level statistics, transmission fluctuations and others) observed in generic quantum chaotic systems. Here we generalize quantum graph theory to the nonlinear domain. The theory will be applied in particular to show the effect of nonlinearity on transmission through networks of nonlinear fibers. Our model may also be used as a simple yet non-trivial model where the universal properties derived from detailed numerical computations of Bose-Einstein condensates in non-regular traps [2][3][4][5][6][7] could be further investigated.Scattering is studied as a stationary process. The main finding is that the sharp resonances which dominate scattering in networks with complex connectivity lead to a dramatic amplification of the nonlinear effects: while non-resonant scattering hardly deviates from the predictions of the linear theory, tuning the parameters to a nearby resonance (without changing the incoming field intensity) brings the system into the nonlinear regime which is signaled by multi-stability and hysteresis. For this reason we revisit the theory of scattering in the linear regime and demonstrate that sharp resonances with large amplification of the incoming wave inside the system are very frequent for graphs compared to other complex (chaotic) scattering systems. The origin of this effect can be related to the topology of the graph (existence of cycles) and leads to a power-law distribution for the amplification. I. THE NONLINEAR SCHRÖDINGER EQUATION ON GRAPHSConsider a general metric graph which consists of V vertices connected by B internal bonds and N leads to infinity as illustrated in Fig. 1. The bonds and leads will be collectively referred to as edges. have the coordinate x l ∈ [0, ∞) and x l = 0 is at the vertex where the lead is attached. The wave function on the graph is a bounded piecewise continuous and differentiable function on the edges written collectively as(1)where ψ e (x e ) is the wave functions on the edge e. While the model can describe far more general settings we restrict ourselves in this exploratory work to the discussion of stationary scattering. The wave function on edge e satisfies the stationary nonlinear Schrödinger equationHere, g e is real nonlinear coupling parameter which we assume constant on each edge (but it may take different values ...

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Section: Implications For Nonlinear Scattering: Multistabilitysupporting

confidence: 90%

Section: B Linear Scattering: Resonances and Amplificationmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Stationary scattering from a nonlinear network

2011

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“…Підставляючи в рівняння (51) компоненти ( , , ) X t x z , одержуємо формулу, що встановлює зв'язок між прямою задачею (1)-( 9) і спряженою задачею ( 23)- (26), що дає можливість отримати явні вирази компонентів градієнта функціонала нев'язки: ( , , )…”

Section: отримання формул виразів ґрадієнтів функціоналу-нев'язкиunclassified

Monte Carlo simulation of dead-end diafiltration of bidispersed particle suspensions

2022

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Схвалено та рекомендовано до друку на засіданні вченої ради Тернопільського національного технічного університету імені Івана Пулюя (протокол №3 від 22 листопада 2022 року) П30 Петрик М.Р. Високопродуктивні методи моделювання та проєктування складних ресурсо-енергозберігаючих процесів у нанопористих і дисперсних середовищах : монографія/ М.Р. Петрик, М.І. Лебовка, І.В. Бойко.-Тернопіль: Вид-во ТНТУ імені Івана Пулюя, 2022.-160 с.Монографія присвячена розробці методів математичного моделювання фізичних та технологічних процесів, що відбуваються нанорозмірних та нанопористих системах різної природи. Розглядаються підходи до програмної реалізації розвинених математичних моделей.Призначена для науковців, аспірантів, що спеціалізуються в галузі математичного моделювання, прикладної математики та інженерії програмного забезпечення.

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Multiple-scale analysis for resonance reflection by a one-dimensional rectangular barrier in the Gross-Pitaevskii problem

Ishkhanyan

Крайнов

2009

Phys. Rev. A

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We consider a quantum above-barrier reflection of a Bose-Einstein condensate by a one-dimensional rectangular potential barrier, or by a potential well, for nonlinear Schrödinger equation ͑Gross-Pitaevskii equation͒ with a small nonlinearity. The most interesting case is realized in resonances when the reflection coefficient is equal to zero for the linear Schrödinger equation. Then the reflection is determined only by small nonlinear term in the Gross-Pitaevskii equation. A simple analytic expression has been obtained for the reflection coefficient produced only by the nonlinearity. An analytical condition is found when common action of potential barrier and nonlinearity produces a zero reflection coefficient. The reflection coefficient is derived analytically in the vicinity of resonances which are shifted by nonlinearity.For studying quantum transmission and reflection, it is the most direct way to find exact solutions of the Schrödinger equation that dominates the dynamics of systems. However, only in a few cases with the simplest potentials, like rectangular well, the Schrödinger equation can be solved exactly. In most circumstances, exact solutions are difficult to obtain due to not only the effect of external field on particles, but also the interaction of particles. The most direct generalization of single-particle case is a tunneling of mean field through a barrier in the Gross-Pitaevskii, or nonlinear Schrödinger equation ͓1,2͔. We emphasize that this is a nonlinear tunneling problem in the mean-field approximation. There have been various theoretical studies. From the theoretical point of view, the main complication in description of a quasistationary scattering process of particles obviously comes from the presence of atom-atom interaction. In leading order, the effect of this interaction is included in a nonlinear term in the Schrödinger-like Gross-Pitaevskii equation for wave function, using the Hartree self-consistent approximation with zero range interaction potential between atoms. The dynamics of solutions of this equation is very complex and rich. The phenomena of instabilities, focusing, and blowup are all concepts related to the nonlinear nature of the systems. Low velocity quantum reflection of Bose-Einstein condensates ͑BEC͒ of ultracold 23 Na atoms from the attractive Casimir-Polder potential of silicon surface was observed experimentally in Refs. ͓3,4͔. The measured reflection probability is in agreement with the theoretical model. Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction was observed in Refs. ͓5,6͔. Their results verify the predicted nonlinear generalization of tunneling oscillations in superconducting and superfluid Josephson junctions for two weakly linked BoseEinstein condensates in a double-well potential. One of the first papers addressing nonlinear resonant tunneling of a BEC has been written by Paul et al. ͓7͔. The most promising results for tunneling experiments are obtained using atomchip-based waveguide interferometry wi...

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Barrier transmission for the nonlinear Schrödinger equation: surprises of nonlinear transport

Cited by 4 publications

References 11 publications

Stationary scattering from a nonlinear network

Stationary scattering from a nonlinear network

Monte Carlo simulation of dead-end diafiltration of bidispersed particle suspensions

Multiple-scale analysis for resonance reflection by a one-dimensional rectangular barrier in the Gross-Pitaevskii problem

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