427 research outputs found
Scaling Properties of Weak Chaos in Nonlinear Disordered Lattices
The Discrete Nonlinear Schroedinger Equation with a random potential in one
dimension is studied as a dynamical system. It is characterized by the length,
the strength of the random potential and by the field density that determines
the effect of nonlinearity. The probability of the system to be regular is
established numerically and found to be a scaling function. This property is
used to calculate the asymptotic properties of the system in regimes beyond our
computational power.Comment: 4 pages, 5 figure
Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread and Singularity
We derive several kinetic equations to model the large scale, low Fresnel
number behavior of the nonlinear Schrodinger (NLS) equation with a rapidly
fluctuating random potential. There are three types of kinetic equations the
longitudinal, the transverse and the longitudinal with friction. For these
nonlinear kinetic equations we address two problems: the rate of dispersion and
the singularity formation.
For the problem of dispersion, we show that the kinetic equations of the
longitudinal type produce the cubic-in-time law, that the transverse type
produce the quadratic-in-time law and that the one with friction produces the
linear-in-time law for the variance prior to any singularity.
For the problem of singularity, we show that the singularity and blow-up
conditions in the transverse case remain the same as those for the homogeneous
NLS equation with critical or supercritical self-focusing nonlinearity, but
they have changed in the longitudinal case and in the frictional case due to
the evolution of the Hamiltonian
Existence and stability of solitons for the nonlinear Schr\"odinger equation on hyperbolic space
We study the existence and stability of ground state solutions or solitons to
a nonlinear stationary equation on hyperbolic space. The method of
concentration compactness applies and shows that the results correlate strongly
to those of Euclidean space.Comment: New: As noted in Banica-Duyckaerts (arXiv:1411.0846), Section 5
should read that for sufficiently large mass, sub-critical problems can be
solved via energy minimization for all d \geq 2 and as a result
Cazenave-Lions results can be applied in Section 6 with the same restriction.
These requirements were addressed by the subsequent work with Metcalfe and
Taylor in arXiv:1203.361
A generalized nonlinear Schr\"odinger equation as model for turbulence, collapse, and inverse cascade
A two-dimensional generalized cubic nonlinear Schr\"odinger equation with
complex coefficients for the group dispersion and nonlinear terms is used to
investigate the evolution of a finite-amplitude localized initial perturbation.
It is found that modulation of the latter can lead to side-band formation, wave
condensation, collapse, turbulence, and inverse cascade, although not all
together nor in that order.Comment: 12 pages, 5 figure
Influence of ion-to-electron temperature ratio on tearing instability and resulting subion-scale turbulence in a low- collisionless plasma
A two-field gyrofluid model including ion finite Larmor radius (FLR)
corrections, magnetic fluctuations along the ambient field and electron inertia
is used to study two-dimensional reconnection in a low collisionless
plasma, in a plane perpendicular to the ambient field. Both moderate and large
values of the ion-to-electron temperature ratio are considered. The
linear growth rate of the tearing instability is computed for various values of
, confirming the convergence to reduced electron magnetodynamics (REMHD)
predictions in the large limit. Comparisons with analytical estimates in
several limit cases are also presented. The nonlinear dynamics leads to a
fully-developed turbulent regime that appears to be sensitive to the value of
the parameter . For , strong large-scale velocity shears
trigger Kelvin-Helmholtz instability, leading to the propagation of the
turbulence through the separatrices, together with the formation of eddies of
size of the order of the electron skin depth. In the regime, the
vortices are significantly smaller and their accurate description requires that
electron FLR effects be taken into account
Distribution of eigenfrequencies for oscillations of the ground state in the Thomas--Fermi limit
In this work, we present a systematic derivation of the distribution of
eigenfrequencies for oscillations of the ground state of a repulsive
Bose-Einstein condensate in the semi-classical (Thomas-Fermi) limit. Our
calculations are performed in 1-, 2- and 3-dimensional settings. Connections
with the earlier work of Stringari, with numerical computations, and with
theoretical expectations for invariant frequencies based on symmetry principles
are also given.Comment: 8 pages, 1 figur
A pulsed atomic soliton laser
It is shown that simultaneously changing the scattering length of an
elongated, harmonically trapped Bose-Einstein condensate from positive to
negative and inverting the axial portion of the trap, so that it becomes
expulsive, results in a train of self-coherent solitonic pulses. Each pulse is
itself a non-dispersive attractive Bose-Einstein condensate that rapidly
self-cools. The axial trap functions as a waveguide. The solitons can be made
robustly stable with the right choice of trap geometry, number of atoms, and
interaction strength. Theoretical and numerical evidence suggests that such a
pulsed atomic soliton laser can be made in present experiments.Comment: 11 pages, 4 figure
Vortices in attractive Bose-Einstein condensates in two dimensions
The form and stability of quantum vortices in Bose-Einstein condensates with
attractive atomic interactions is elucidated. They appear as ring bright
solitons, and are a generalization of the Townes soliton to nonzero winding
number . An infinite sequence of radially excited stationary states appear
for each value of , which are characterized by concentric matter-wave rings
separated by nodes, in contrast to repulsive condensates, where no such set of
states exists. It is shown that robustly stable as well as unstable regimes may
be achieved in confined geometries, thereby suggesting that vortices and their
radial excited states can be observed in experiments on attractive condensates
in two dimensions.Comment: 4 pages, 3 figure
Nonlinearity Management in Higher Dimensions
In the present short communication, we revisit nonlinearity management of the
time-periodic nonlinear Schrodinger equation and the related averaging
procedure. We prove that the averaged nonlinear Schrodinger equation does not
support the blow-up of solutions in higher dimensions, independently of the
strength in the nonlinearity coefficient variance. This conclusion agrees with
earlier works in the case of strong nonlinearity management but contradicts
those in the case of weak nonlinearity management. The apparent discrepancy is
explained by the divergence of the averaging procedure in the limit of weak
nonlinearity management.Comment: 9 pages, 1 figure
Dispersion and collapse of wave maps
We study numerically the Cauchy problem for equivariant wave maps from 3+1
Minkowski spacetime into the 3-sphere. On the basis of numerical evidence
combined with stability analysis of self-similar solutions we formulate two
conjectures. The first conjecture states that singularities which are produced
in the evolution of sufficiently large initial data are approached in a
universal manner given by the profile of a stable self-similar solution. The
second conjecture states that the codimension-one stable manifold of a
self-similar solution with exactly one instability determines the threshold of
singularity formation for a large class of initial data. Our results can be
considered as a toy-model for some aspects of the critical behavior in
formation of black holes.Comment: 14 pages, Latex, 9 eps figures included, typos correcte
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