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

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 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

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

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

Transition to miscibility in linearly coupled binary dipolar Bose-Einstein condensates

We investigate effects of dipole-dipole (DD) interactions on immiscibility-miscibility transitions (IMTs) in two-component Bose-Einstein condensates (BECs) trapped in the harmonic-oscillator (HO) potential, with the components linearly coupled by a resonant electromagnetic field (accordingly, the components represent two different spin states of the same atom). The problem is studied by means of direct numerical simulations. Different mutual orientations of the dipolar moments in the two components are considered. It is shown that, in the binary BEC formed by dipoles with the same orientation and equal magnitudes, the IMT cannot be induced by the DD interaction alone, being possible only in the presence of the linear coupling between the components, while the miscibility threshold is affected by the DD interactions. However, in the binary condensate with the two dipolar components polarized in opposite directions, the IMT can be induced \emph{without} any linear coupling. Further, we demonstrate that those miscible and immiscible localized states, formed in the presence of the DD interactions, which are unstable evolve into robust breathers, which tend to keep the original miscibility or immiscibility, respectively. An exception is the case of a very strong DD attraction, when narrow stationary modes are destroyed by the instability. The binary BEC composed of co-polarized dipoles with different magnitudes are briefly considered too.Comment: 10 figure

Models of spin-orbit coupled oligomers

We address the stability and dynamics of eigenmodes in linearly-shaped strings (dimers, trimers, tetramers, and pentamers) built of droplets of a binary Bose-Einstein condensate (BEC). The binary BEC is composed of atoms in two pseudo-spin states with attractive interactions, dressed by properly arranged laser fields, which induce the (pseudo-) spin-orbit (SO) coupling. We demonstrate that the SO-coupling terms help to create eigenmodes of particular types in the strings. Dimer, trimer, and pentamer eigenmodes of the linear system, which correspond to the zero eigenvalue (EV, alias chemical potential) extend into the nonlinear ones, keeping an exact analytical form, while tetramers do not admit such a continuation, because the respective spectrum does not contain a zero EV. Stability areas of these modes shrink with the increasing nonlinearity. Besides these modes, other types of nonlinear states, which are produced by the continuation of their linear counterparts corresponding to some nonzero EVs, are found in a numerical form (including ones for the tetramer system). They are stable in nearly entire existence regions in trimer and pentamer systems, but only in a very small area for the tetramers. Similar results are also obtained, but not displayed in detail, for hexa- and septamers.Comment: Chaos, in pres