3,705 research outputs found
High frequency integrable regimes in nonlocal nonlinear optics
We consider an integrable model which describes light beams propagating in
nonlocal nonlinear media of Cole-Cole type. The model is derived as high
frequency limit of both Maxwell equations and the nonlocal nonlinear
Schroedinger equation. We demonstrate that for a general form of nonlinearity
there exist selfguided light beams. In high frequency limit nonlocal
perturbations can be seen as a class of phase deformation along one direction.
We study in detail nonlocal perturbations described by the dispersionless
Veselov-Novikov (dVN) hierarchy. The dVN hierarchy is analyzed by the reduction
method based on symmetry constraints and by the quasiclassical Dbar-dressing
method. Quasiclassical Dbar-dressing method reveals a connection between
nonlocal nonlinear geometric optics and the theory of quasiconformal mappings
of the plane.Comment: 45 pages, 4 figure
Paraxial light in a Cole-Cole nonlocal medium: integrable regimes and singularities
Nonlocal nonlinear Schroedinger-type equation is derived as a model to
describe paraxial light propagation in nonlinear media with different `degrees'
of nonlocality. High frequency limit of this equation is studied under specific
assumptions of Cole-Cole dispersion law and a slow dependence along propagating
direction. Phase equations are integrable and they correspond to dispersionless
limit of Veselov-Novikov hierarchy. Analysis of compatibility among intensity
law (dependence of intensity on the refractive index) and high frequency limit
of Poynting vector conservation law reveals the existence of singular
wavefronts. It is shown that beams features depend critically on the
orientation properties of quasiconformal mappings of the plane. Another class
of wavefronts, whatever is intensity law, is provided by harmonic minimal
surfaces. Illustrative example is given by helicoid surface. Compatibility with
first and third degree nonlocal perturbations and explicit solutions are also
discussed.Comment: 12 pages, 2 figures; eq. (36) corrected, minor change
Target Patterns in a 2-D Array of Oscillators with Nonlocal Coupling
We analyze the effect of adding a weak, localized, inhomogeneity to a two
dimensional array of oscillators with nonlocal coupling. We propose and also
justify a model for the phase dynamics in this system. Our model is a
generalization of a viscous eikonal equation that is known to describe the
phase modulation of traveling waves in reaction-diffusion systems. We show the
existence of a branch of target pattern solutions that bifurcates from the
spatially homogeneous state when , the strength of the
inhomogeneity, is nonzero and we also show that these target patterns have an
asymptotic wavenumber that is small beyond all orders in .
The strategy of our proof is to pose a good ansatz for an approximate form of
the solution and use the implicit function theorem to prove the existence of a
solution in its vicinity. The analysis presents two challenges. First, the
linearization about the homogeneous state is a convolution operator of
diffusive type and hence not invertible on the usual Sobolev spaces. Second, a
regular perturbation expansion in does not provide a good ansatz
for applying the implicit function theorem since the nonlinearities play a
major role in determining the relevant approximation, which also needs to be
"correct" to all orders in . We overcome these two points by
proving Fredholm properties for the linearization in appropriate Kondratiev
spaces and using a refined ansatz for the approximate solution, which obtained
using matched asymptotics.Comment: 39 pages, 1 figur
A scalar nonlocal bifurcation of solitary waves for coupled nonlinear Schroedinger systems
An explanation is given for previous numerical results which suggest a
certain bifurcation of `vector solitons' from scalar (single-component)
solitary waves in coupled nonlinear Schroedinger (NLS) systems. The bifurcation
in question is nonlocal in the sense that the vector soliton does not have a
small-amplitude component, but instead approaches a solitary wave of one
component with two infinitely far-separated waves in the other component. Yet,
it is argued that this highly nonlocal event can be predicted from a purely
local analysis of the central solitary wave alone. Specifically the
linearisation around the central wave should contain asymptotics which grow at
precisely the speed of the other-component solitary waves on the two wings.
This approximate argument is supported by both a detailed analysis based on
matched asymptotic expansions, and numerical experiments on two example
systems. The first is the usual coupled NLS system involving an arbitrary ratio
between the self-phase and cross-phase modulation terms, and the second is a
coupled NLS system with saturable nonlinearity that has recently been
demonstrated to support stable multi-peaked solitary waves. The asymptotic
analysis further reveals that when the curves which define the proposed
criterion for scalar nonlocal bifurcations intersect with boundaries of certain
local bifurcations, the nonlocal bifurcation could turn from scalar to
non-scalar at the intersection. This phenomenon is observed in the first
example. Lastly, we have also selectively tested the linear stability of
several solitary waves just born out of scalar nonlocal bifurcations. We found
that they are linearly unstable. However, they can lead to stable solitary
waves through parameter continuation.Comment: To appear in Nonlinearit
Azimuthal Modulational Instability of Vortices in the Nonlinear Schr\"odinger Equation
We study the azimuthal modulational instability of vortices with different
topological charges, in the focusing two-dimensional nonlinear Schr{\"o}dinger
(NLS) equation. The method of studying the stability relies on freezing the
radial direction in the Lagrangian functional of the NLS in order to form a
quasi-one-dimensional azimuthal equation of motion, and then applying a
stability analysis in Fourier space of the azimuthal modes. We formulate
predictions of growth rates of individual modes and find that vortices are
unstable below a critical azimuthal wave number. Steady state vortex solutions
are found by first using a variational approach to obtain an asymptotic
analytical ansatz, and then using it as an initial condition to a numerical
optimization routine. The stability analysis predictions are corroborated by
direct numerical simulations of the NLS. We briefly show how to extend the
method to encompass nonlocal nonlinearities that tend to stabilize solutions.Comment: 8 pages, 6 figures, in press for Optics Communication
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