264 research outputs found
Fundamental solitons in discrete lattices with a delayed nonlinear response
The formation of unstaggered localized modes in dynamical lattices can be
supported by the interplay of discreteness and nonlinearity with a finite
relaxation time. In rapidly responding nonlinear media, on-site discrete
solitons are stable, and their broad inter-site counterparts are marginally
stable, featuring a virtually vanishing real instability eigenvalue. The
solitons become unstable in the case of the slowly relaxing nonlinearity. The
character of the instability alters with the increase of the delay time, which
leads to a change in the dynamics of unstable discrete solitons. They form
robust localized breathers in rapidly relaxing media, and decay into
oscillatory diffractive pattern in the lattices with a slow nonlinear response.
Marginally stable solitons can freely move across the lattice.Comment: 8 figure
Discrete localized modes supported by an inhomogeneous defocusing nonlinearity
We report that infinite and semi-infinite lattices with spatially
inhomogeneous self-defocusing (SDF)\ onsite nonlinearity, whose strength
increases rapidly enough toward the lattice periphery, support stable
unstaggered (UnST) discrete bright solitons, which do not exist in lattices
with the spatially uniform SDF nonlinearity. The UnST solitons coexist with
stable staggered (ST) localized modes, which are always possible under the
defocusing onsite nonlinearity. The results are obtained in a numerical form,
and also by means of variational approximation (VA). In the semi-infinite
(truncated) system, some solutions for the UnST surface solitons are produced
in an exact form. On the contrary to surface discrete solitons in uniform
truncated lattices, the threshold value of the norm vanishes for the UnST
solitons in the present system. Stability regions for the novel UnST solitons
are identified. The same results imply the existence of ST discrete solitons in
lattices with the spatially growing self-focusing nonlinearity, where such
solitons cannot exist either if the nonlinearity is homogeneous. In addition, a
lattice with the uniform onsite SDF nonlinearity and exponentially decaying
inter-site coupling is introduced and briefly considered too. Via a similar
mechanism, it may also support UnST discrete solitons, under the action of the
SDF nonlinearity. The results may be realized in arrayed optical waveguides and
collisionally inhomogeneous Bose-Einstein condensates trapped in deep optical
lattices. A generalization for a two-dimensional system is briefly considered
too.Comment: 14 pages, 7 figures, accepted for publication in PR
Interface solitons in one-dimensional locally-coupled lattice systems
Fundamental solitons pinned to the interface between two discrete lattices
coupled at a single site are investigated. Serially and parallel-coupled
identical chains (\textit{System 1} and \textit{System 2}), with the
self-attractive on-site cubic nonlinearity, are considered in one dimension. In
these two systems, which can be readily implemented as arrays of nonlinear
optical waveguides, symmetric, antisymmetric and asymmetric solitons are
investigated by means of the variational approximation (VA) and numerical
methods. The VA demonstrates that the antisymmetric solitons exist in the
entire parameter space, while the symmetric and asymmetric modes can be found
below some critical value of the coupling parameter. Numerical results confirm
these predictions for the symmetric and asymmetric fundamental modes. The
existence region of numerically found antisymmetric solitons is also limited by
a certain value of the coupling parameter. The symmetric solitons are
destabilized via a supercritical symmetry-breaking pitchfork bifurcation, which
gives rise to stable asymmetric solitons, in both systems. The antisymmetric
fundamental solitons, which may be stable or not, do not undergo any
bifurcation. In bistability regions stable antisymmetric solitons coexist with
either symmetric or asymmetric ones.Comment: 9 figure
Localized modes in mini-gaps opened by periodically modulated intersite coupling in two-dimensional nonlinear lattices
Spatially periodic modulation of the intersite coupling in two-dimensional
(2D) nonlinear lattices modifies the eigenvalue spectrum by opening mini-gaps
in it. This work aims to build stable localized modes in the new bandgaps.
Numerical analysis shows that single-peak and composite two- and four-peak
discrete static solitons and breathers emerge as such modes in certain
parameter areas inside the mini-gaps of the 2D superlattice induced by the
periodic modulation of the intersite coupling along both directions.The
single-peak solitons and four-peak discrete solitons are stable in a part of
their existence domain, while unstable stationary states (in particular,
two-soliton complexes) may readily transform into robust localized breathers.Comment: Chaos, in pres
Extreme Events in Nonlinear Lattices
The spatiotemporal complexity induced by perturbed initial excitations
through the development of modulational instability in nonlinear lattices with
or without disorder, may lead to the formation of very high amplitude,
localized transient structures that can be named as extreme events. We analyze
the statistics of the appearance of these collective events in two different
universal lattice models; a one-dimensional nonlinear model that interpolates
between the integrable Ablowitz-Ladik (AL) equation and the nonintegrable
discrete nonlinear Schr\"odinger (DNLS) equation, and a two-dimensional
disordered DNLS equation. In both cases, extreme events arise in the form of
discrete rogue waves as a result of nonlinear interaction and rapid coalescence
between mobile discrete breathers. In the former model, we find power-law
dependence of the wave amplitude distribution and significant probability for
the appearance of extreme events close to the integrable limit. In the latter
model, more importantly, we find a transition in the the return time
probability of extreme events from exponential to power-law regime. Weak
nonlinearity and moderate levels of disorder, corresponding to weak chaos
regime, favour the appearance of extreme events in that case.Comment: Invited Chapter in a Special Volume, World Scientific. 19 pages, 9
figure
Nonlinear symmetry breaking of Aharonov-Bohm cages
We study the influence of mean field cubic nonlinearity on Aharonov-Bohm
caging in a diamond lattice with synthetic magnetic flux. For sufficiently weak
nonlinearities the Aharonov-Bohm caging persists as periodic nonlinear
breathing dynamics. Above a critical nonlinearity, symmetry breaking induces a
sharp transition in the dynamics and enables stronger wavepacket spreading.
This transition is distinct from other flatband networks, where continuous
spreading is induced by effective nonlinear hopping or resonances with
delocalized modes, and is in contrast to the quantum limit, where two-particle
hopping enables arbitrarily large spreading. This nonlinear symmetry breaking
transition is readily observable in femtosecond laser-written waveguide arrays.Comment: 6 pages, 5 figure
High- and low-frequency phonon modes in dipolar quantum gases trapped in deep lattices
We study normal modes propagating on top of the stable uniform background in
arrays of dipolar Bose-Einstein condensate (BEC) droplets trapped in a deep
optical lattice. Both the on-site mean-field dynamics of the droplets and their
displacement due to the repulsive dipole-dipole interactions (DDIs) are taken
into account. Dispersion relations for two modes, \textit{viz}., high- and low-
frequency counterparts of optical and acoustic phonon modes in condensed
matter, are derived analytically and verified by direct simulations, for both
cases of the repulsive and attractive contact interactions. The (counterpart of
the) optical-phonon branch does not exist without the DDIs. These results are
relevant in the connection to emerging experimental techniques enabling
real-time imaging of the condensate dynamics and direct experimental
measurement of phonon dispersion relations in BECs.Comment: Physical Review A, in pres
Nonlinear localized flatband modes with spin-orbit coupling
We report the coexistence and properties of stable compact localized states
(CLSs) and discrete solitons (DSs) for nonlinear spinor waves on a flatband
network with spin-orbit coupling (SOC). The system can be implemented by means
of a binary Bose-Einstein condensate loaded in the corresponding optical
lattice. In the linear limit, the SOC opens a minigap between flat and
dispersive bands in the system's bandgap structure, and preserves the existence
of CLSs at the flatband frequency, simultaneously lowering their symmetry.
Adding onsite cubic nonlinearity, the CLSs persist and remain available in an
exact analytical form, with frequencies which are smoothly tuned into the
minigap. Inside of the minigap, the CLS and DS families are stable in narrow
areas adjacent to the FB. Deep inside the semi-infinite gap, both the CLSs and
DSs are stable too.Comment: 10 figures, Physical Review B, in pres
Soliton stability and collapse in the discrete nonpolynomial Schrodinger equation with dipole-dipole interactions
The stability and collapse of fundamental unstaggered bright solitons in the
discrete Schrodinger equation with the nonpolynomial on-site nonlinearity,
which models a nearly one-dimensional Bose-Einstein condensate trapped in a
deep optical lattice, are studied in the presence of the long-range
dipole-dipole (DD) interactions. The cases of both attractive and repulsive
contact and DD interaction are considered. The results are summarized in the
form of stability/collapse diagrams in the parametric space of the model, which
demonstrate that the the attractive DD interactions stabilize the solitons and
help to prevent the collapse. Mobility of the discrete solitons is briefly
considered too.Comment: 6 figure
Discrete solitons in an array of quantum dots
We develop a theory for the interaction of classical light fields with an a
chain of coupled quantum dots (QDs), in the strong-coupling regime, taking into
account the local-field effects. The QD chain is modeled by a one-dimensional
(1D) periodic array of two-level quantum particles with tunnel coupling between
adjacent ones. The local-field effect is taken into regard as QD depolarization
in the Hartree-Fock-Bogoliubov approximation. The dynamics of the chain is
described by a system of two discrete nonlinear Schr\"{o}dinger (DNLS)
equations for local amplitudes of the probabilities of the ground and first
excited states. The two equations are coupled by a cross-phase-modulation cubic
terms, produced by the local-field action, and by linear terms too. In
comparison with previously studied DNLS systems, an essentially new feature is
a phase shift between the intersite-hopping constants in the two equations. By
means of numerical solutions, we demonstrate that, in this QD chain, Rabi
oscillations (RO) self-trap into stable bright\textit{\ Rabi solitons} or
\textit{Rabi breathers}. Mobility of the solitons is considered too. The
related behavior of observable quantities, such as energy, inversion, and
electric-current density, is given a physical interpretation. The results apply
to a realistic region of physical parameters.Comment: 12 pages, 10 figures, Phys. Rev. B, in pres
- …