45 research outputs found
Localised nonlinear modes in the PT-symmetric double-delta well Gross-Pitaevskii equation
We construct exact localised solutions of the PT-symmetric Gross-Pitaevskii
equation with an attractive cubic nonlinearity. The trapping potential has the
form of two -function wells, where one well loses particles while the
other one is fed with atoms at an equal rate. The parameters of the constructed
solutions are expressible in terms of the roots of a system of two
transcendental algebraic equations. We also furnish a simple analytical
treatment of the linear Schr\"odinger equation with the PT-symmetric
double- potential.Comment: To appear in Proceedings of the 15 Conference on Pseudo-Hermitian
Hamiltonians in Quantum Physics, May 18-23 2015, Palermo, Italy (Springer
Proceedings in Physics, 2016
Discrete solitons in PT-symmetric lattices
We prove existence of discrete solitons in infinite parity-time (PT-)
symmetric lattices by means of analytical continuation from the anticontinuum
limit. The energy balance between dissipation and gain implies that in the
anticontinuum limit the solitons are constructed from elementary PT-symmetric
blocks such as dimers, quadrimers, or more general oligomers. We consider in
detail a chain of coupled dimers, analyze bifurcations of discrete solitons
from the anticontinuum limit and show that the solitons are stable in a
sufficiently large region of the lattice parameters. The generalization of the
approach is illustrated on two examples of networks of quadrimers, for which
stable discrete solitons are also found.Comment: 6 pages, 6 figures; accepted to EPL, www.epletters.ne
Stability of localized modes in PT-symmetric nonlinear potentials
We report on detailed investigation of the stability of localized modes in
the nonlinear Schrodinger equations with a nonlinear parity-time (alias PT)
symmetric potential. We are particularly focusing on the case where the
spatially-dependent nonlinearity is purely imaginary. We compute the Evans
function of the linear operator determining the linear stability of localized
modes. Results of the Evans function analysis predict that for sufficiently
small dissipation localized modes become stable when the propagation constant
exceeds certain threshold value. This is the case for periodic and
-shaped complex potentials where the modes having widths comparable with
or smaller than the characteristic width of the complex potential are stable,
while broad modes are unstable. In contrast, in complex potentials that change
linearly with transverse coordinate all modes are stable, what suggests that
the relation between width of the modes and spatial size of the complex
potential define the stability in the general case. These results were
confirmed using the direct propagation of the solutions for the mentioned
examples.Comment: 6 pages, 4 figures; accepted to Europhysics Letters,
https://www.epletters.net
Eigenvalues bifurcating from the continuum in two-dimensional potentials generating non-Hermitian gauge fields
It has been recently shown that complex two-dimensional (2D) potentials
can be used to emulate
non-Hermitian matrix gauge fields in optical waveguides. Here and are
the transverse coordinates, and are real functions,
is a small parameter, and is the imaginary unit.
The real potential is required to have at least two discrete eigenvalues
in the corresponding 1D Schr\"odinger operator. When both transverse directions
are taken into account, these eigenvalues become thresholds embedded in the
continuous spectrum of the 2D operator. Small nonzero corresponds
to a non-Hermitian perturbation which can result in a bifurcation of each
threshold into an eigenvalue. Accurate analysis of these eigenvalues is
important for understanding the behavior and stability of optical waves
propagating in the artificial non-Hermitian gauge potential. Bifurcations of
complex eigenvalues out of the continuum is the main object of the present
study. We obtain simple asymptotic expansions in that describe
the behavior of bifurcating eigenvalues. The lowest threshold can bifurcate
into a single eigenvalue, while every other threshold can bifurcate into a pair
of complex eigenvalues. These bifurcations can be controlled by the Fourier
transform of function evaluated at certain isolated points of the
reciprocal space. When the bifurcation does not occur, the continuous spectrum
of 2D operator contains a quasi-bound-state which is characterized by a
strongly localized central peak coupled to small-amplitude but nondecaying
tails. The analysis is applied to the case examples of parabolic and
double-well potentials . In the latter case, the bifurcation of complex
eigenvalues can be dampened if the two wells are widely separated.Comment: 24 pages, 6 figures; accepted for Annals of Physic
Bright and dark solitons in the systems with strong light-matter coupling: exact solutions and numerical simulations
We theoretically study bright and dark solitons in an experimentally relevant
hybrid system characterized by strong light-matter coupling. We find that the
corresponding two-component model supports a variety of coexisting moving
solitons including bright solitons on zero and nonzero background, dark-gray
and gray-gray dark solitons. The solutions are found in the analytical form by
reducing the two-component problem to a single stationary equation with
cubic-quintic nonlinearity. All found solutions coexist under the same set of
the model parameters, but, in a properly defined linear limit, approach
different branches of the polariton dispersion relation for linear waves.
Bright solitons with zero background feature an oscillatory-instability
threshold which can be associated with a resonance between the edges of the
continuous spectrum branches. `Half-topological' dark-gray and nontopological
gray-gray solitons are stable in wide parametric ranges below the modulational
instability threshold, while bright solitons on the constant-amplitude pedestal
are unstable.Comment: 11 pages, 11 figures; accepted for Phys. Rev.