Stability of multiple pulses in discrete systems

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.

Non-standard Hubbard models in optical lattices: a review

Originally, the Hubbard model was derived for describing the behavior of strongly correlated electrons in solids. However, for over a decade now, variations of it have also routinely been implemented with ultracold atoms in optical lattices, allowing their study in a clean, essentially defect-free environment. Here, we review some of the vast literature on this subject, with a focus on more recent non-standard forms of the Hubbard model. After giving an introduction to standard (fermionic and bosonic) Hubbard models, we discuss briefly common models for mixtures, as well as the so-called extended Bose-Hubbard models, that include interactions between neighboring sites, next-neighbor sites, and so on. The main part of the review discusses the importance of additional terms appearing when refining the tight-binding approximation for the original physical Hamiltonian. Even when restricting the models to the lowest Bloch band is justified, the standard approach neglects the density-induced tunneling (which has the same origin as the usual on-site interaction). The importance of these contributions is discussed for both contact and dipolar interactions. For sufficiently strong interactions, the effects related to higher Bloch bands also become important even for deep optical lattices. Different approaches that aim at incorporating these effects, mainly via dressing the basis, Wannier functions with interactions, leading to effective, density-dependent Hubbard-type models, are reviewed. We discuss also examples of Hubbard-like models that explicitly involve higher p orbitals, as well as models that dynamically couple spin and orbital degrees of freedom. Finally, we review mean-field nonlinear Schrödinger models of the Salerno type that share with the non-standard Hubbard models nonlinear coupling between the adjacent sites. In that part, discrete solitons are the main subject of consideration. We conclude by listing some open problems, to be addressed in the future.

Modulational instability and breathing motion in the two-dimensional nonlinear Schrödinger equation with a one-dimensional harmonic potential

Modulational instability and breathing motion are studied in the two-dimensional nonlinear Schrödinger (NLS) equation trapped by the one-dimensional harmonic potential. The trapping potential is uniform in the y direction and the wave function is confined in the x direction. A breathing motion appears when the initial condition is close to a stationary solution which is uniform in the y direction. The amplitude of the breathing motion is larger in the two-dimensional system than that in the corresponding one-dimensional system. Coupled equations of the one-dimensional NLS equation and two variational parameters are derived by the variational approximation to understand the amplification of the breathing motion qualitatively. On the other hand, there is a breathing solution in the x direction which is uniform in the y direction to the two-dimensional NLS equation. It is shown that the modulational instability along the y direction is suppressed when the breathing motion is sufficiently strong, even if the norm is above the critical value of the collapse.

Unstaggered-staggered solitons in two-component discrete nonlinear Schrödinger lattices

We present stable bright solitons built of coupled unstaggered and staggered components in a symmetric system of two discrete nonlinear Schrödinger equations with the attractive self-phase-modulation nonlinearity, coupled by the repulsive cross-phase-modulation interaction. These mixed modes are of a "symbiotic" type, as each component in isolation may only carry ordinary unstaggered solitons. The results are obtained in an analytical form, using the variational and Thomas-Fermi approximations (VA and TFA), and the generalized Vakhitov-Kolokolov (VK) criterion for the evaluation of the stability. The analytical predictions are verified against numerical results. Almost all the symbiotic solitons are predicted by the VA quite accurately and are stable. Close to a boundary of the existence region of the solitons (which may feature several connected branches), there are broad solitons which are not well approximated by the VA and are unstable.

Soliton complexity in the damped-driven nonlinear Schrödinger equation: stationary to periodic to quasiperiodic complexes

Stationary and oscillatory bound states, or complexes, of the damped-driven solitons are numerically path-followed in the parameter space. We compile a chart of the two-soliton attractors, complementing the one-soliton attractor chart.

Two-dimensional paradigm for symmetry breaking: the nonlinear Schrödinger equation with a four-well potential

We study the existence and stability of localized modes in the two-dimensional (2D) nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation with a symmetric four-well potential. Using the corresponding four-mode approximation, we trace the parametric evolution of the trapped stationary modes, starting from the linear limit, and thus derive a complete bifurcation diagram for families of the stationary modes. This provides the picture of spontaneous symmetry breaking in the fundamental 2D setting. In a broad parameter region, the predictions based on the four-mode decomposition are found to be in good agreement with full numerical solutions of the NLS/GP equation. Stability properties of the stationary states coincide with those suggested by the corresponding discrete model in the large-amplitude limit. The dynamics of unstable modes is explored by means of direct simulations. Finally, in addition to the full analysis for the case of the self-attractive nonlinearity, the bifurcation diagram for the case of self-repulsion is briefly considered too.

Discrete solitons and vortices in hexagonal and honeycomb lattices: existence, stability, and dynamics

We consider a prototypical dynamical lattice model, namely, the discrete nonlinear Schrödinger equation on nonsquare lattice geometries. We present a systematic classification of the solutions that arise in principal six-lattice-site and three-lattice-site contours in the form of both discrete multipole solitons and discrete vortices. Additionally to identifying the possible states, we analytically track their linear stability both qualitatively and quantitatively. We find that among the six-site configurations, the "hexapole" of alternating phases (0-pi) , as well as the vortex of topological charge S=2 have intervals of stability; among three-site states, only the vortex of topological charge S=1 may be stable in the case of focusing nonlinearity. These conclusions are confirmed both for hexagonal and for honeycomb lattices by means of detailed numerical bifurcation analysis of the stationary states from the anticontinuum limit, and by direct simulations to monitor the dynamical instabilities, when the latter arise. The dynamics reveal a wealth of nonlinear behavior resulting not only in single-site solitary wave forms, but also in robust multisite breathing structures.

Three-dimensional vortex solitons in quasi-two-dimensional lattices

We consider the three-dimensional (3D) Gross-Pitaevskii or nonlinear Schrödinger equation with a quasi-2D square-lattice potential (which corresponds to the optical lattice trapping a self-attractive Bose-Einstein condensate, or, in some approximation, to a photonic-crystal fiber, in terms of nonlinear optics). Stable 3D solitons, with embedded vorticity S=1 and 2, are found by means of the variational approximation and in a numerical form. They are built, basically, as sets of four fundamental solitons forming a rhombus, with phase shifts piS2 between adjacent sites, and an empty site in the middle. The results demonstrate two species of stable 3D solitons, which were not studied before, viz., localized vortices ("spinning light bullets," in terms of optics) with S>1 , and vortex solitons (with any S not equal 0 ) supported by a lattice in the 3D space. Typical scenarios of instability development (collapse or decay) of unstable localized vortices are identified too.

X , Y , and Z waves: extended structures in nonlinear lattices

We propose a new type of waveforms in two-dimensional (2D) and three-dimensional (3D) discrete media-multilegged extended nonlinear structures (ENSs), built as arrays of lattice solitons (tiles and stones, in the 2D and 3D cases, respectively). We study the stability of the tiles and stones analytically, and then extend them numerically to complete ENS forms for both 2D and 3D lattices, aiming to single out stable ENSs. The predicted patterns can be realized in Bose-Einstein condensates trapped in deep optical lattices, crystals built of microresonators, and 2D photonic crystals. In the latter case, the patterns provide for a technique for writing reconfigurable virtual partitions in multipurpose photonic devices.

Interaction-induced localization of anomalously diffracting nonlinear waves

We study experimentally the interactions between normal solitons and tilted beams in glass waveguide arrays. We find that as a tilted beam, traversing away from a normally propagating soliton, coincides with the self-defocusing regime of the array, it can be refocused and routed back into any of the intermediate sites due to the interaction, as a function of the initial phase difference. Numerically, distinct parameter regimes exhibiting this behavior of the interaction are identified.

Discrete solitons and vortices in the two-dimensional Salerno model with competing nonlinearities

An anisotropic lattice model in two spatial dimensions, with on-site and intersite cubic nonlinearities (the Salerno model), is introduced, with emphasis on the case in which the intersite nonlinearity is self-defocusing, competing with on-site self-focusing. The model applies, for example, to a dipolar Bose-Einstein condensate trapped in a deep two-dimensional (2D) optical lattice. Soliton families of two kinds are found in the model: ordinary ones and cuspons, with peakons at the border between them. Stability borders for the ordinary solitons are found, while all cuspons (and peakons) are stable. The Vakhitov-Kolokolov criterion does not apply to cuspons, but for the ordinary solitons it correctly identifies the stability limits. In direct simulations, unstable solitons evolve into localized pulsons. Varying the anisotropy parameter, we trace a transition between the solitons in 1D and 2D versions of the model. In the isotropic model, we also construct discrete vortices of two types, on-site and intersite centered (vortex crosses and squares, respectively), and identify their stability regions. In simulations, unstable vortices in the noncompeting model transform into regular solitons, while in the model with the competing nonlinearities they evolve into localized vortical pulsons, which maintain their topological character. Bound states of regular solitons and vortices are constructed too, and their stability is identified.

Solitons in the Salerno model with competing nonlinearities

We consider a lattice equation (Salerno model) combining onsite self-focusing and intersite self-defocusing cubic terms, which may describe a Bose-Einstein condensate of dipolar atoms trapped in a strong periodic potential. In the continuum approximation, the model gives rise to solitons in a finite band of frequencies, with sechlike solitons near one edge, and an exact peakon solution at the other. A similar family of solitons is found in the discrete system, including a peakon; beyond the peakon, the family continues in the form of cuspons. Stability of the lattice solitons is explored through computation of eigenvalues for small perturbations, and by direct simulations. A small part of the family is unstable (in that case, the discrete solitons transform into robust pulsonic excitations); both peakons and cuspons are stable. The Vakhitov-Kolokolov criterion precisely explains the stability of regular solitons and peakons, but does not apply to cuspons. In-phase and out-of-phase bound states of solitons are also constructed. They exchange their stability at a point where the bound solitons are peakons. Mobile solitons, composed of a moving core and background, exist up to a critical value of the strength of the self-defocusing intersite nonlinearity. Colliding solitons always merge into a single pulse.

Matter-wave solitons in nonlinear optical lattices

We introduce a dynamical model of a Bose-Einstein condensate based on the one-dimensional (1D) Gross-Pitaevskii equation (GPE) with a nonlinear optical lattice (NOL), which is represented by the cubic term whose coefficient is periodically modulated in the coordinate. The model describes a situation when the atomic scattering length is spatially modulated, via the optically controlled Feshbach resonance, in an optical lattice created by interference of two laser beams. Relatively narrow solitons supported by the NOL are predicted by means of the variational approximation (VA), and an averaging method is applied to broad solitons. A different feature is a minimum norm (number of atoms), N = N(min), necessary for the existence of solitons. The VA predicts N(min) very accurately. Numerical results are chiefly presented for the NOL with the zero spatial average value of the nonlinearity coefficient. Solitons with values of the amplitude A larger than at N = N(min) are stable. Unstable solitons with smaller, but not too small, A rearrange themselves into persistent breathers. For still smaller A, the soliton slowly decays into radiation without forming a breather. Broad solitons with very small A are practically stable, as their decay is extremely slow. These broad solitons may freely move across the lattice, featuring quasielastic collisions. Narrow solitons, which are strongly pinned to the NOL, can easily form stable complexes. Finally, the weakly unstable low-amplitude solitons are stabilized if a cubic term with a constant coefficient, corresponding to weak attraction, is included in the GPE.

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.

Three-dimensional nonlinear lattices: from oblique vortices and octupoles to discrete diamonds and vortex cubes

We construct a variety of novel localized topological structures in the 3D discrete nonlinear Schrödinger equation. The states can be created in Bose-Einstein condensates trapped in strong optical lattices and crystals built of microresonators. These new structures, most of which have no counterparts in lower dimensions, range from multipole patterns and diagonal vortices to vortex "cubes" (stack of two quasiplanar vortices) and "diamonds" (formed by two orthogonal vortices).

Breather lattice and its stabilization for the modified Korteweg-de Vries equation

We obtain an exact solution for the breather lattice solution of the modified Korteweg-de Vries equation. Numerical simulation of the breather lattice demonstrates its instability due to the breather-breather interaction. However, such multibreather structures can be stabilized through the concurrent application of ac driving and viscous damping terms.

Discrete vector solitons in two-dimensional nonlinear waveguide arrays: solutions, stability, and dynamics

We identify and investigate bimodal (vector) solitons in models of square-lattice arrays of nonlinear optical waveguides. These vector self-localized states are, in fact, self-induced channels in a nonlinear photonic-crystal matrix. Such two-dimensional discrete vector solitons are possible in waveguide arrays in which each element carries two light beams that are either orthogonally polarized or have different carrier wavelengths. Estimates of the physical parameters necessary to support such soliton solutions in waveguide arrays are given. Using Newton relaxation methods, we obtain stationary vector-soliton solutions, and examine their stability through the computation of linearized eigenvalues for small perturbations. Our results may also be applicable to other systems such as two-component Bose-Einstein condensates trapped in a two-dimensional optical lattice.

Instabilities and bifurcations of nonlinear impurity modes

We study the structure and stability of nonlinear impurity modes in the discrete nonlinear Schrödinger equation with a single on-site nonlinear impurity emphasizing the effects of interplay between discreteness, nonlinearity, and disorder. We show how the interaction of a nonlinear localized mode (a discrete soliton or discrete breather) with a repulsive impurity generates a family of stationary states near the impurity site, as well as examine both theoretical and numerical criteria for the transition between different localized states via a cascade of bifurcations.

Continuum approach to discreteness

We study analytically and numerically continuum models derived on the basis of Padé approximations and their effectiveness in modeling spatially discrete systems. We not only analyze features of the temporal dynamics that can be captured through these continuum approaches (e.g., shape oscillations, radiation effects, and trapping) but also point out ones that cannot be captured (such as Peierls-Nabarro barriers and Bloch oscillations). We analyze the role of such methods in providing an effective "homogenization" of spatially discrete, as well as of heterogeneous continuum equations. Finally, we develop numerical methods for solving such equations and use them to establish the range of validity of these continuum approximations, as well as to compare them with other semicontinuum approximations.

Spatial optical solitons in nonlinear photonic crystals

We study spatial optical solitons in a one-dimensional nonlinear photonic crystal created by an array of thin-film nonlinear waveguides, the so-called Dirac-comb nonlinear lattice. We analyze modulational instability of the extended Bloch-wave modes and also investigate the existence and stability of bright, dark, and "twisted" spatially localized modes in such periodic structures. Additionally, we discuss both similarities and differences of our general results with the simplified models of nonlinear periodic media described by the discrete nonlinear Schrödinger equation, derived in the tight-binding approximation, and the coupled-mode theory, valid for shallow periodic modulations of the optical refractive index.

Energy of internal modes of nonlinear waves and complex frequencies due to symmetry breaking

Considering a class of Hamiltonian systems, it is demonstrated that energy of the internal modes with real frequencies supported by nonlinear waves and appearing due to perturbations breaking a continuous symmetry has its sign determined by the symmetry itself, independently of the nature of the perturbations. In particular, it is shown that negative energy modes emerge as a result of the breaking of the phase symmetry in the perturbed nonlinear-Schrödinger equation. An expression for energy of the Vakhitov-Kolokolov internal modes is also derived. Comparative analysis of the energy signs of the internal modes in these two cases explains the ubiquity of instabilities with complex frequencies of solitary and continuous waves in systems with broken phase symmetry.

Multipulses in discrete Hamiltonian nonlinear systems

In this work, the behavior of multipulses in discrete Hamiltonian nonlinear systems is investigated. The discrete nonlinear Schrödinger equation is used as the benchmark system for this study. A singular perturbation methodology as well as a variational approach are implemented in order to identify the dominant factors in the discrete problem. The results of the two methodologies are shown to coincide in assessing the interplay of discreteness and exponential tail-tail pulse interaction. They also allow one to understand why, contrary to what is believed for their continuum siblings, discrete systems can sustain (static) multipulse configurations, a conclusion that is subsequently verified by numerical experiment.

Discrete vortex solitons

Localized states in the discrete two-dimensional (2D) nonlinear Schrödinger equation is found: vortex solitons with an integer vorticity S. While Hamiltonian lattices do not conserve angular momentum or the topological invariant related to it, we demonstrate that the soliton's vorticity may be conserved as a dynamical invariant. Linear stability analysis and direct simulations concur in showing that fundamental vortex solitons, with S=1, are stable if the intersite coupling C is smaller than some critical value C((1))(cr). At C>C((1))(cr), an instability sets in through a quartet of complex eigenvalues appearing in the linearized equations. Direct simulations reveal that an unstable vortex soliton with S=1 first splits into two usual solitons with S=0 (in accordance with the prediction of the linear analysis), but then an instability-induced spontaneous symmetry breaking takes place: one of the secondary solitons with S=0 decays into radiation, while the other one survives. We demonstrate that the usual (S=0) 2D solitons in the model become unstable, at C>C((0))(cr) approximately 2.46C((1))(cr), in a different way, via a pair of imaginary eigenvalues omega which bifurcate into instability through omega=0. Except for the lower-energy S=1 solitons that are centered on a site, we also construct ones which are centered between lattice sites which, however, have higher energy than the former. Vortex solitons with S=2 are found too, but they are always unstable. Solitons with S=1 and S=0 can form stable bound states.