Dark solitons in discrete lattices

Dynamics of discrete solitons in the fractional discrete nonlinear Schrödinger equation with the quasi-Riesz derivative

We elaborate a fractional discrete nonlinear Schrödinger (FDNLS) equation based on an appropriately modified definition of the Riesz fractional derivative, which is characterized by its Lévy index (LI). This FDNLS equation represents a novel discrete system, in which the nearest-neighbor coupling is combined with long-range interactions, that decay as the inverse square of the separation between lattice sites. The system may be realized as an array of parallel quasi-one-dimensional Bose-Einstein condensates composed of atoms or small molecules carrying, respectively, a permanent magnetic or electric dipole moment. The dispersion relation (DR) for lattice waves and the corresponding propagation band in the system's linear spectrum are found in an exact form for all values of LI. The DR is consistent with the continuum limit, differing in the range of wave numbers. Formation of single-site and two-site discrete solitons is explored, starting from the anticontinuum limit and continuing the analysis in the numerical form up to the existence boundary of the discrete solitons. Stability of the solitons is identified in terms of eigenvalues for small perturbations, and verified in direct simulations. Mobility of the discrete solitons is considered too, by means of an estimate of the system's Peierls-Nabarro potential barrier, and with the help of direct simulations. Collisions between persistently moving discrete solitons are also studied.

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

Supersolid gap soliton in a Bose-Einstein condensate and optical ring cavity coupling system

The system of a transversely pumped Bose-Einstein condensate (BEC) coupled to a lossy ring cavity can favor a supersolid steady state. Here we find the existence of supersolid gap soliton in such a driven-dissipative system. By numerically solving the mean-field atom-cavity field coupling equations, gap solitons of a few different families have been identified. Their dynamical properties, including stability, propagation, and soliton collision, are also studied. Due to the feedback atom-intracavity field interaction, these supersolid gap solitons show numerous new features compared with the usual BEC gap solitons in static optical lattices.

Algebraic bright and vortex solitons in defocusing media

We demonstrate that spatially inhomogeneous defocusing nonlinear landscapes with the nonlinearity coefficient growing toward the periphery as (1+|r|(α)) support one- and two-dimensional fundamental and higher-order bright solitons, as well as vortex solitons, with algebraically decaying tails. The energy flow of the solitons converges as long as nonlinearity growth rate exceeds the dimensionality, i.e., α>D. Fundamental solitons are always stable, while multipoles and vortices are stable if the nonlinearity growth rate is large enough.

Discrete solitons in electromechanical resonators

We consider a particular type of parametrically driven discrete Klein-Gordon system describing microdevices and nanodevices, with integrated electrical and mechanical functionality. Using a multiscale expansion method we reduce the system to a discrete nonlinear Schrödinger equation. Analytical and numerical calculations are performed to determine the existence and stability of fundamental bright and dark discrete solitons admitted by the Klein-Gordon system through the discrete Schrödinger equation. We show that a parametric driving can not only destabilize onsite bright solitons, but also stabilize intersite bright discrete solitons and onsite and intersite dark solitons. Most importantly, we show that there is a range of values of the driving coefficient for which dark solitons are stable, for any value of the coupling constant, i.e., oscillatory instabilities are totally suppressed. Stability windows of all the fundamental solitons are presented and approximations to the onset of instability are derived using perturbation theory, with accompanying numerical results. Numerical integrations of the Klein-Gordon equation are performed, confirming the relevance of our analysis.

Interlaced linear-nonlinear optical waveguide arrays

The system of coupled discrete equations describing a two-component superlattice with interlaced linear and nonlinear constituents is studied as a basis for investigating binary waveguide arrays, such as ribbed AlGaAs structures, among others. Compared to the single nonlinear lattice, the interlaced system exhibits an extra band-gap controlled by the, suitably chosen by design, relative detuning. In more general physics settings, this system represents a discretization scheme for the single-equation-based continuous models in media with transversely modulated linear and nonlinear properties. Continuous wave solutions and the associated modulational instability are fully analytically investigated and numerically tested for focusing and defocusing nonlinearity. The propagation dynamics and the stability of periodic modes are also analytically investigated for the case of zero Bloch momentum. In the band-gaps a variety of stable discrete solitary modes, dipole or otherwise, in-phase or of staggered type are found and discussed.

Dark soliton dynamics and interactions in continuous-wave-induced lattices

The dynamics of dark spatial soliton beams and their interaction under the presence of a continuous wave (CW), which dynamically induces a photonic lattice, are investigated. It is shown that appropriate selection of the characteristic parameters of the CW result in controllable steering of a single soliton as well as controllable interaction between two solitons. Depending on the CW parameters, the soliton angle of propagation can be changed drastically, while two-soliton interaction can be either enhanced or reduced, suggesting a reconfigurable soliton control mechanism. Our analytical approach, based on the variational perturbation method, provides a dynamical system for the dark soliton evolution parameters. Analytical results are shown in good agreement with direct numerical simulations.

Dark solitons in discrete lattices: saturable versus cubic nonlinearities

In the present work, we study dark solitons in dynamical lattices with the saturable nonlinearity and compare them to those in lattices with the cubic nonlinearity. This comparison has become especially relevant in light of recent experimental developments in the former context. The stability properties of the fundamental waves, for both onsite and intersite modes, are examined analytically and corroborated by numerical results. Our findings indicate that for both models onsite solutions are stable for sufficiently small values of the coupling between adjacent nodes, while intersite solutions are always unstable. The nature of the instability (which is oscillatory for onsite solutions at large coupling and exponential for inter-site solutions) is probed via the dynamical evolution of unstable solitary waves through appropriately crafted numerical experiments; typically, these computations result in dynamic motion of the originally stationary solitary waves. Another key finding, consistent with recent experimental results, is that the instability growth rate for the saturable nonlinearity is found to be smaller than that of the cubic case.

Formation and light guiding properties of dark solitons in one-dimensional waveguide arrays

We report on the formation of dark discrete solitons in a nonlinear periodic system consisting of evanescently coupled channel waveguides in defocusing lithium niobate. Localized nonlinear dark modes displaying a phase jump in the center that is located either on-channel (mode A) or in-between channels (mode B) are formed, which is to our knowledge the first experimental observation of mode B. By numerical simulations we find that the saturable nature of the nonlinearity is responsible for the improved stability of mode B. The ability of the induced refractive index structures to guide light of a low-power probe beam is demonstrated.

Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear Kronig-Penney model

A novel method for obtaining analytical solitary wave solutions of the nonlinear Kronig-Penney model in periodic photonic structures with self-defocusing nonlinearity is applied for providing generic families of solutions corresponding to the gaps of the linear band structure. Characteristic cases are shown to be quite robust under propagation.

Discrete vector solitons in one-dimensional lattices in photorefractive media

We construct families of two-component spatial solitons in a one-dimensional lattice with saturable on-site nonlinearity (focusing or defocusing) in a photorefractive crystal. We identify 14 species of vector solitons, depending on their type (bright/dark), phase (in-phase/staggered), and location on the lattice (on/off-site). Two species of the bright/bright type form entirely stable soliton families, four species are partially stable (depending on the value of the propagation constant), while the remaining eight species are completely unstable. "Symbiotic" soliton pairs (of the bright/dark type), which contain components that cannot exist in isolation in the same model, are found as well.

Discrete dark solitons with multiple holes

We consider staggered dark solitons admitted by the discrete nonlinear Schrödinger equation with focusing cubic nonlinearity. In particular, we focus on the study of dark solitons with several holes characterized by the number of zeros in the uncoupled case. Such structures reveal interesting behaviors, such as stable intersite dark solitons. All of the structures have no counterpart in the strong coupling limit since they disappear in a saddle-node bifurcation. We also consider the evolution of structures with multiple holes representing an interaction between multiple dark solitons in a very discrete case.

Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation

In this paper we study numerically existence and stability of exact dark waves on the (nonintegrable) discrete nonlinear Schrödinger equation for a finite one-dimensional lattice. These are solutions that bifurcate from stationary dark modes with constant background intensity and zero intensity at a site, and whose initial state translates exactly one site each period of the internal oscillations. We show that exact dark waves are characterized by an oscillatory background whose wavelength is closely related with the velocity. Faster dark waves require smaller wavelengths. For slow enough velocity dark waves are linearly stable, but when trying to continue numerically a solution towards higher velocities bifurcations appear, due to rearrangements in the oscillatory tail in order to make possible a decreasing of the wavelength. However, in principle, one might control the stability of an exact dark wave adjusting a phase factor which plays the role of a discreteness parameter. In addition, we also study the regimes of existence and stability for stationary discrete gray modes, which are exact solutions with phase-twisted constant-amplitude background and nonzero minimum intensity. Also such solutions develop envelope oscillations on top of the homogeneous background when continued into moving phase-twisted solutions.

Discretizing light behaviour in linear and nonlinear waveguide lattices

Light propagating in linear and nonlinear waveguide lattices exhibits behaviour characteristic of that encountered in discrete systems. The diffraction properties of these systems can be engineered, which opens up new possibilities for controlling the flow of light that would have been otherwise impossible in the bulk: these effects can be exploited to achieve diffraction-free propagation and minimize the power requirements for nonlinear processes. In two-dimensional networks of waveguides, self-localized states--or discrete solitons--can travel along 'wire-like' paths and can be routed to any destination port. Such possibilities may be useful for photonic switching architectures.

Discrete solitons in nonlinear zigzag optical waveguide arrays with tailored diffraction properties

We show that the discrete diffraction properties of a nonlinear optical zigzag waveguide array can be significantly modified, by exploiting the topological arrangement of the lattice itself. This introduces extended interactions (beyond nearest neighbors), which, in turn, affect the lattice dispersion relation within the Brillouin zone. As a result of this band alteration, we demonstrate that altogether different families of discrete soliton solutions are possible, which are stable over a wide range of parameters. In the regime where instabilities occur, all scenarios are considered in detail. By appropriately engineering the geometrical configuration of the array we find both standing and traveling diffraction-free beams. Our method opens opportunities for diffraction management that can be employed to generate low-power spatial discrete optical solitons.

Self-focusing and defocusing in waveguide arrays

We show that two regimes of diffraction exist in arrays of waveguides, depending upon the input conditions. At higher powers, normal diffraction leads to self-focusing and to the formation of bright solitons through the nonlinear Kerr effect. By slightly changing the input conditions, light experiences anomalous diffraction and is nonlinearly defocused. For the first time, self-focusing and self-defocusing have been achieved for the same medium, structure, and wavelength.

Stability of multiple pulses in discrete systems

The stability of multiple-pulse solutions to the discrete nonlinear Schrödinger equation is considered. A bound state of widely separated single pulses is rigorously shown to be unstable, unless the phase shift Delta phi between adjacent pulses satisfies Delta phi=pi. This instability is accounted for by positive real eigenvalues in the linearized system. The analysis leading to the instability result does not, however, determine the linear stability of those multiple pulses for which Delta phi=pi between adjacent pulses. A direct variational approach for a two-pulse predicts that it is linearly stable if Delta phi=pi, and if the separation between the individual pulses satisfies a certain condition. The variational approach can easily be generalized to study the stability of N pulses for any N>or=3. The analysis is supplemented with a detailed numerical stability analysis.

Diffraction management

By using the diffraction properties of waveguide arrays, we propose a scheme to produce structures with designed diffraction. We fabricated arrays with reduced, canceled, and even reversed diffraction. Results of experiments with such waveguides are presented and compared with the predictions made by coupled-mode theory.