Discrete nonlinear Schrödinger equation dynamics in complex networks

Perakis, Fivos; Tsironis, G. P.

doi:10.1016/j.physleta.2010.11.053

Cited by 13 publications

(22 citation statements)

References 19 publications

(38 reference statements)

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“…Numerical results indicate that the combination of structural disorder with nonlinearity leads to almost complete localization of the wavepacket, for values bellow the self-trapping threshold (χ c = 7.5), which is in agreement with previous investigations [25,26]. It would be very illuminating to perform the experiments discussed here and actually see whether one can control diffusion through the lattice geometry (disorder) and laser intensity (nonlinearity), and probe experimentally the small-world regime.…”

Section: Discussionsupporting

confidence: 89%

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Small-world networks of optical fiber lattices

Perakis

Mattheakis

Tsironis

2014

J. Opt.

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We use a simple dynamical model and explore coherent dynamics of wavepackets in complex networks of optical fibers. We start from a symmetric lattice and through the application of a Monte-Carlo criterion we introduce structural disorder and deform the lattice into a small-world network regime. We investigate in the latter both structural (correlation length) as well as dynamical (diffusion exponent) properties and find that both exhibit a rapid crossover from the ordered to the fully random regime. For a critical value of the structural disorder parameter ρ ≈ 0.25 transport changes from ballistic to sub-diffusive due to the creation strongly connected local clusters and channels of preferential transport in the small world regime.

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Section: Discussionsupporting

confidence: 89%

“…To simulate the dynamics of the excited wavepacket on the 2D lattice use the discrete nonlinear Schrödinger equation (DNLS) [22,24,25,32]:…”

Section: Dynamicsmentioning

confidence: 99%

Small-world networks of optical fiber lattices

Perakis

Mattheakis

Tsironis

2014

J. Opt.

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“…a result that agrees with the linear solutions derived in ref. [1]. We note that in the linear case the system executes oscillations between p = 1 and p lin,…”

Section: A Nonlinear Mean Field Limitmentioning

confidence: 89%

“…Nonlinear dynamics in complex networks incorporates competition of propagation, nonlinearity and bond disorder and may find some applications in complex natural or manmade materials. For a given random network of N sites the limit of fully coupled lattice where each site is connected to every other one with the same strength plays an important role since, for zero nonlinearity, introduces a high degree of degeneracy and thus localization [1]. It is therefore interesting to probe the dynamics in this fully coupled limit when the network is nonlinear.…”

Section: Introductionmentioning

confidence: 99%

Exact dynamics for fully connected nonlinear networks

Tsironis

2011

Physics Letters A

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We investigate the dynamics of the discrete nonlinear Schr\"{o}dinger equation in fully connected networks. For a localized initial condition the exact solution shows the existence of two dynamical transitions as a function of the nonlinearity parameter, a hyperbolic and a trigonometric one. In the latter the network behaves exactly as the corresponding linear one but with a renormalized frequency.Comment: 6 pages, 2 figure

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“…Equation (14) can also be expressed in real space, by replacing Eq. (7) in Eq. (14), obtaining This results are confirmed by direct computations of cases (a), (b) and (c), using 'exact' Eq.…”

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confidence: 99%

Diffusion in infinite and semi-infinite lattices with long-range coupling

Martínez¹,

Molina²

2012

J. Phys. A: Math. Theor.

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We prove that for a one-dimensional infinite lattice, with long-range coupling among sites, the diffusion of an initial delta-like pulse in the bulk, is ballistic at all times. We obtain a closedform expression for the mean square displacement (MSD) as a function of time, and show some cases including finite range coupling, exponentially decreasing coupling and power-law decreasing coupling. For the case of an initial excitation at the edge of the lattice, we find an approximate expression for the MSD that predicts ballistic behavior at long times, in agreement with numerical results.PACS numbers: 63.10.+a, 66.30.-h The physics of discrete systems have been a topic of interest for many years, because it can give rise to completely different phenomenology in comparison with that present in homogeneous continuous systems. In particular, discrete periodic systems are found in many different contexts including condensed matter physics, optics, Bose-Einstein condensates and magnetic metamaterials among others. Under the appropriate approximation, they can all be described by some variant of the discrete Schrödinger (DS) equation [1]. In that way, many of these systems display the phenomenology common to periodic systems such as the presence of a band structure, discrete diffraction, Bloch oscillations, dynamic localization, Zener tunneling, to name a few.Usually, the DS equation is used in the weak-coupling limit, where the interaction among sites includes nearestneighbors only. This is a good approximation in cases where the coupling among sites decays very quickly with distance, like the exponentially-decreasing coupling found in optical waveguide arrays. However, there are cases where it is advisable to go beyond this approximation. An example of that is a split-ring resonator (SRR) array, where the interaction among the basic units is dipolar in nature and therefore, the coupling decreases as the inverse cubic power of the mutual distance.When coupling beyond nearest-neighbors are considered, the number of possible routes of energy exchange increases. The dynamical evolution of excited pulses in finite 1D and 2D lattices with anisotropic couplings and up to second nearest-neighbor couplings, has been explored in [2] by means of the Green function formalism. Experimental observation of the influence of second order coupling in linear and nonlinear optical zig-zag waveguide arrays has been recently carried out [3]. In a different context, a recent work [4], shows that long-range coupling in low dimensional system can induce a phase transition from delocalized to localized modes. Another interesting scenario that can be modeled as a discrete system with long-range couplings is that of complex networks [5], where the distances between nodes are not necessarily physical.In this Letter we carry out an analytical and numerical study on the diffusion of an initially localized pulse propagating in a one-dimensional discrete periodic lattice, in the presence of arbitrary long-range couplings. We focus on two cases of in...

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Discrete nonlinear Schrödinger equation dynamics in complex networks

Cited by 13 publications

References 19 publications

Small-world networks of optical fiber lattices

Small-world networks of optical fiber lattices

Exact dynamics for fully connected nonlinear networks

Diffusion in infinite and semi-infinite lattices with long-range coupling

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