158 research outputs found
Collapse and stable self-trapping for Bose-Einstein condensates with 1/r^b type attractive interatomic interaction potential
We consider dynamics of Bose-Einstein condensates with long-range attractive
interaction proportional to and arbitrary angular dependence. It is
shown exactly that collapse of Bose-Einstein condensate without contact
interactions is possible only for . Case is critical and requires
number of particles to exceed critical value to allow collapse. Critical
collapse in that case is strong one trapping into collapsing region a finite
number of particles.
Case is supercritical with expected weak collapse which traps rapidly
decreasing number of particles during approach to collapse. For
singularity at is not strong enough to allow collapse but attractive
interaction admits stable self-trapping even in absence of external
trapping potential
Collapse in the nonlocal nonlinear Schr\"odinger equation
We discuss spatial dynamics and collapse scenarios of localized waves
governed by the nonlinear Schr\"{o}dinger equation with nonlocal nonlinearity.
Firstly, we prove that for arbitrary nonsingular attractive nonlocal nonlinear
interaction in arbitrary dimension collapse does not occur. Then we study in
detail the effect of singular nonlocal kernels in arbitrary dimension using
both, Lyapunoff's method and virial identities. We find that for for a
one-dimensional case, i.e. for , collapse cannot happen for nonlocal
nonlinearity. On the other hand, for spatial dimension and singular
kernel , no collapse takes place if , whereas
collapse is possible if . Self-similar solutions allow us to find
an expression for the critical distance (or time) at which collapse should
occur in the particular case of kernels. Moreover, different
evolution scenarios for the three dimensional physically relevant case of Bose
Einstein condensate are studied numerically for both, the ground state and a
higher order toroidal state with and without an additional local repulsive
nonlinear interaction. In particular, we show that presence of an additional
local repulsive term can prevent collapse in those cases
Strong Collapse Turbulence in Quintic Nonlinear Schr\"odinger Equation
We consider the quintic one dimensional nonlinear Schr\"odinger equation with
forcing and both linear and nonlinear dissipation. Quintic nonlinearity results
in multiple collapse events randomly distributed in space and time forming
forced turbulence. Without dissipation each of these collapses produces finite
time singularity but dissipative terms prevents actual formation of
singularity. In statistical steady state of the developed turbulence the
spatial correlation function has a universal form with the correlation length
determined by the modulational instability scale. The amplitude fluctuations at
that scale are nearly-Gaussian while the large amplitude tail of probability
density function (PDF) is strongly non-Gaussian with power-like behavior. The
small amplitude nearly-Gaussian fluctuations seed formation of large collapse
events. The universal spatio-temporal form of these events together with the
PDF for their maximum amplitudes define the power-like tail of PDF for large
amplitude fluctuations, i.e., the intermittency of strong turbulence.Comment: 14 pages, 17 figure
Collapse of solitary waves near transition from supercritical to subcritical bifurcations
We study both analytically and numerically the nonlinear stage of the
instability of one-dimensional solitons in a small vicinity of the transition
point from supercritical to subcritical bifurcations in the framework of the
generalized nonlinear Schr\"{o}dinger equation. It is shown that near the
collapsing time the pulse amplitude and its width demonstrate the self-similar
behavior with a small asymmetry at the pulse tails due to self-steepening. This
theory is applied to both solitary interfacial deep-water waves and envelope
water waves with a finite depth and short optical pulses in fibers as well
Ultrashort filaments of light in weakly-ionized, optically-transparent media
Modern laser sources nowadays deliver ultrashort light pulses reaching few
cycles in duration, high energies beyond the Joule level and peak powers
exceeding several terawatt (TW). When such pulses propagate through
optically-transparent media, they first self-focus in space and grow in
intensity, until they generate a tenuous plasma by photo-ionization. For free
electron densities and beam intensities below their breakdown limits, these
pulses evolve as self-guided objects, resulting from successive equilibria
between the Kerr focusing process, the chromatic dispersion of the medium, and
the defocusing action of the electron plasma. Discovered one decade ago, this
self-channeling mechanism reveals a new physics, widely extending the frontiers
of nonlinear optics. Implications include long-distance propagation of TW beams
in the atmosphere, supercontinuum emission, pulse shortening as well as
high-order harmonic generation. This review presents the landmarks of the
10-odd-year progress in this field. Particular emphasis is laid to the
theoretical modeling of the propagation equations, whose physical ingredients
are discussed from numerical simulations. Differences between femtosecond
pulses propagating in gaseous or condensed materials are underlined. Attention
is also paid to the multifilamentation instability of broad, powerful beams,
breaking up the energy distribution into small-scale cells along the optical
path. The robustness of the resulting filaments in adverse weathers, their
large conical emission exploited for multipollutant remote sensing, nonlinear
spectroscopy, and the possibility to guide electric discharges in air are
finally addressed on the basis of experimental results.Comment: 50 pages, 38 figure
Emergent complex neural dynamics
A large repertoire of spatiotemporal activity patterns in the brain is the
basis for adaptive behaviour. Understanding the mechanism by which the brain's
hundred billion neurons and hundred trillion synapses manage to produce such a
range of cortical configurations in a flexible manner remains a fundamental
problem in neuroscience. One plausible solution is the involvement of universal
mechanisms of emergent complex phenomena evident in dynamical systems poised
near a critical point of a second-order phase transition. We review recent
theoretical and empirical results supporting the notion that the brain is
naturally poised near criticality, as well as its implications for better
understanding of the brain
A novel brain partition highlights the modular skeleton shared by structure and function
Elucidating the intricate relationship between brain structure and function, both in healthy and pathological conditions, is a key challenge for modern neuroscience. Recent progress in neuroimaging has helped advance our understanding of this important issue, with diffusion images providing information about structural connectivity (SC) and functional magnetic resonance imaging shedding light on resting state functional connectivity (rsFC). Here, we adopt a systems approach, relying on modular hierarchical clustering, to study together SC and rsFC datasets gathered independently from healthy human subjects. Our novel approach allows us to find a common skeleton shared by structure and function from which a new, optimal, brain partition can be extracted. We describe the emerging common structure-function modules (SFMs) in detail and compare them with commonly employed anatomical or functional parcellations. Our results underline the strong correspondence between brain structure and resting-state dynamics as well as the emerging coherent organization of the human brain.Work supported by Ikerbasque: The Basque Foundation for Science, Euskampus at UPV/EHU, Gobierno Vasco (Saiotek SAIO13-PE13BF001) and Junta de Andalucía (P09-FQM-4682) to JMC; Ikerbasque Visiting Professor to SS; Junta de Andalucía (P09-FQM-4682) and Spanish Ministry of Economy and Competitiveness (FIS2013-43201-P) to MAM; the European Union’s Seventh Framework Programme (ICT-FET FP7/2007-2013, FET Young Explorers scheme) under grant agreement n. 284772 BRAIN BOW (www.brainbowproject.eu) and by the Joint Italy—Israel Laboratory on Neuroscience to PB. For results validation (figure S8), data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University
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