198 research outputs found
Pinned modes in two-dimensional lossy lattices with local gain and nonlinearity
We introduce a system with one or two amplified nonlinear sites ("hot spots",
HSs) embedded into a two-dimensional linear lossy lattice. The system describes
an array of evanescently coupled optical or plasmonic waveguides, with gain
applied at selected HS cores. The subject of the analysis is discrete solitons
pinned to the HSs. The shape of the localized modes is found in
quasi-analytical and numerical forms, using a truncated lattice for the
analytical consideration. Stability eigenvalues are computed numerically, and
the results are supplemented by direct numerical simulations. In the case of
self-focusing nonlinearity, the modes pinned to a single HS are stable or
unstable when the nonlinearity includes the cubic loss or gain, respectively.
If the nonlinearity is self-defocusing, the unsaturated cubic gain acting at
the HS supports stable modes in a small parametric area, while weak cubic loss
gives rise to a bistability of the discrete solitons. Symmetric and
antisymmetric modes pinned to a symmetric set of two HSs are considered too.Comment: Philosophical Transactions of the Royal Society A, in press (a
special issue on "Localized structures in dissipative media"
An explicit unconditionally stable numerical method for solving damped nonlinear Schr\"{o}dinger equations with a focusing nonlinearity
This paper introduces an extension of the time-splitting sine-spectral (TSSP)
method for solving damped focusing nonlinear Schr\"{o}dinger equations (NLS).
The method is explicit, unconditionally stable and time transversal invariant.
Moreover, it preserves the exact decay rate for the normalization of the wave
function if linear damping terms are added to the NLS. Extensive numerical
tests are presented for cubic focusing nonlinear Schr\"{o}dinger equations in
2d with a linear, cubic or a quintic damping term. Our numerical results show
that quintic or cubic damping always arrests blowup, while linear damping can
arrest blowup only when the damping parameter \dt is larger than a threshold
value \dt_{\rm th}. We note that our method can also be applied to solve the
3d Gross-Pitaevskii equation with a quintic damping term to model the dynamics
of a collapsing and exploding Bose-Einstein condensate (BEC).Comment: SIAM Journal on Numerical Analysis, to appea
Stable topological modes in two-dimensional Ginzburg-Landau models with trapping potentials
Complex Ginzburg-Landau (CGL) models of laser media (with the cubic-quintic
nonlinearity) do not contain an effective diffusion term, which makes all
vortex solitons unstable in these models. Recently, it has been demonstrated
that the addition of a two-dimensional periodic potential, which may be induced
by a transverse grating in the laser cavity, to the CGL equation stabilizes
compound (four-peak) vortices, but the most fundamental "crater-shaped"
vortices (CSVs), alias vortex rings, which are, essentially, squeezed into a
single cell of the potential, have not been found before in a stable form. In
this work we report families of stable compact CSVs with vorticity S=1 in the
CGL model with the external potential of two different types: an axisymmetric
parabolic trap, and the periodic potential. In both cases, we identify
stability region for the CSVs and for the fundamental solitons (S=0). Those
CSVs which are unstable in the axisymmetric potential break up into robust
dipoles. All the vortices with S=2 are unstable, splitting into tripoles.
Stability regions for the dipoles and tripoles are identified too. The periodic
potential cannot stabilize CSVs with S>=2 either; instead, families of stable
compact square-shaped quadrupoles are found
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