285 research outputs found
Statistical Mechanics of Charged Particles in the Pressure of Magnetic Irregularities
Statistical mechanics of charged particles in presence of magnetic irregularitie
Ring-type singular solutions of the biharmonic nonlinear Schrodinger equation
We present new singular solutions of the biharmonic nonlinear Schrodinger
equation in dimension d and nonlinearity exponent 2\sigma+1. These solutions
collapse with the quasi self-similar ring profile, with ring width L(t) that
vanishes at singularity, and radius proportional to L^\alpha, where
\alpha=(4-\sigma)/(\sigma(d-1)). The blowup rate of these solutions is
1/(3+\alpha) for 4/d\le\sigma<4, and slightly faster than 1/4 for \sigma=4.
These solutions are analogous to the ring-type solutions of the nonlinear
Schrodinger equation.Comment: 21 pages, 13 figures, research articl
Non-Gaussian Statistics of Multiple Filamentation
We consider the statistics of light amplitude fluctuations for the
propagation of a laser beam subjected to multiple filamentation in an amplified
Kerr media, with both linear and nonlinear dissipation. Dissipation arrests the
catastrophic collapse of filaments, causing their disintegration into almost
linear waves. These waves form a nearly-Gaussian random field which seeds new
filaments. For small amplitudes the probability density function (PDF) of light
amplitude is close to Gaussian, while for large amplitudes the PDF has a long
power-like tail which corresponds to strong non-Gaussian fluctuations, i.e.
intermittency of strong optical turbulence. This tail is determined by the
universal form of near singular filaments and the PDF for the maximum
amplitudes of the filaments
Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons
We present a unified approach for qualitative and quantitative analysis of
stability and instability dynamics of positive bright solitons in
multi-dimensional focusing nonlinear media with a potential (lattice), which
can be periodic, periodic with defects, quasiperiodic, single waveguide, etc.
We show that when the soliton is unstable, the type of instability dynamic that
develops depends on which of two stability conditions is violated.
Specifically, violation of the slope condition leads to an amplitude
instability, whereas violation of the spectral condition leads to a drift
instability. We also present a quantitative approach that allows to predict the
stability and instability strength
Generalized Neighbor-Interaction Models Induced by Nonlinear Lattices
It is shown that the tight-binding approximation of the nonlinear
Schr\"odinger equation with a periodic linear potential and periodic in space
nonlinearity coefficient gives rise to a number of nonlinear lattices with
complex, both linear and nonlinear, neighbor interactions. The obtained
lattices present non-standard possibilities, among which we mention a
quasi-linear regime, where the pulse dynamics obeys essentially the linear
Schr{\"o}dinger equation. We analyze the properties of such models both in
connection with their modulational stability, as well as in regard to the
existence and stability of their localized solitary wave solutions
Formation of singularities for equivariant 2+1 dimensional wave maps into the two-sphere
In this paper we report on numerical studies of the Cauchy problem for
equivariant wave maps from 2+1 dimensional Minkowski spacetime into the
two-sphere. Our results provide strong evidence for the conjecture that large
energy initial data develop singularities in finite time and that singularity
formation has the universal form of adiabatic shrinking of the degree-one
harmonic map from into .Comment: 14 pages, 5 figures, final version to be published in Nonlinearit
Continuations of the nonlinear Schr\"odinger equation beyond the singularity
We present four continuations of the critical nonlinear \schro equation (NLS)
beyond the singularity: 1) a sub-threshold power continuation, 2) a
shrinking-hole continuation for ring-type solutions, 3) a vanishing
nonlinear-damping continuation, and 4) a complex Ginzburg-Landau (CGL)
continuation. Using asymptotic analysis, we explicitly calculate the limiting
solutions beyond the singularity. These calculations show that for generic
initial data that leads to a loglog collapse, the sub-threshold power limit is
a Bourgain-Wang solution, both before and after the singularity, and the
vanishing nonlinear-damping and CGL limits are a loglog solution before the
singularity, and have an infinite-velocity{\rev{expanding core}} after the
singularity. Our results suggest that all NLS continuations share the universal
feature that after the singularity time , the phase of the singular core
is only determined up to multiplication by . As a result,
interactions between post-collapse beams (filaments) become chaotic. We also
show that when the continuation model leads to a point singularity and
preserves the NLS invariance under the transformation and
, the singular core of the weak solution is symmetric
with respect to . Therefore, the sub-threshold power and
the{\rev{shrinking}}-hole continuations are symmetric with respect to ,
but continuations which are based on perturbations of the NLS equation are
generically asymmetric
Self-similar solutions and collective coordinate methods for Nonlinear Schrodinger Equations
In this paper we study the phase of self-similar solutions to general
Nonlinear Schr\"odinger equations. From this analysis we gain insight on the
dynamics of nontrivial solutions and a deeper understanding of the way
collective coordinate methods work. We also find general evolution equations
for the most relevant dynamical parameter corresponding to the width of
the solution. These equations are exact for self-similar solutions and provide
a shortcut to find approximate evolution equations for the width of
non-self-similar solutions similar to those of collective coordinate methods
Azimuthally polarized spatial dark solitons: exact solutions of Maxwell's equations in a Kerr medium
Spatial Kerr solitons, typically associated with the standard paraxial
nonlinear Schroedinger equation, are shown to exist to all nonparaxial orders,
as exact solutions of Maxwell's equations in the presence of vectorial Kerr
effect. More precisely, we prove the existence of azimuthally polarized,
spatial, dark soliton solutions of Maxwell's equations, while exact linearly
polarized (2+1)-D solitons do not exist. Our ab initio approach predicts the
existence of dark solitons up to an upper value of the maximum field amplitude,
corresponding to a minimum soliton width of about one fourth of the wavelength.Comment: 4 pages, 4 figure
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