Creation and Characterization of Matter-Wave Breathers.

We report the creation of quasi-1D excited matter-wave solitons, "breathers," by quenching the strength of the interactions in a Bose-Einstein condensate with attractive interactions. We characterize the resulting breathing dynamics and quantify the effects of the aspect ratio of the confining potential, the strength of the quench, and the proximity of the 1D-3D crossover for the two-soliton breather. Furthermore, we demonstrate the complex dynamics of a three-soliton breather created by a stronger interaction quench. Our experimental results, which compare well with numerical simulations, provide a pathway for utilizing matter-wave breathers to explore quantum effects in large many-body systems.

Observation of incoherently coupled dark-bright vector solitons in single-mode fibers.

We report experimental observation of incoherently coupled dark-bright vector solitons in single-mode fibers. Properties of the vector solitons accord well with those predicted by the respective systems of incoherently coupled nonlinear Schrödinger equations. To our knowledge, this is the first experimental observation of temporal incoherently coupled dark-bright solitons in single-mode fibers.

Nonlinearity-induced localization in a periodically driven semidiscrete system.

We demonstrate that nonlinearity plays a constructive role in supporting the robustness of dynamical localization in a system which is discrete in one dimension and continuous in the orthogonal one. In the linear regime, time-periodic modulation of the gradient strength along the discrete axis leads to the usual rapid spread of an initially confined wave packet. Addition of the cubic nonlinearity makes the dynamics drastically different, inducing robust localization of moving wave packets. Similar nonlinearity-induced effects are also produced in the presence of a combination of static and oscillating linear potentials. The predicted dynamical localization in the nonlinear medium can be realized in photonic lattices and Bose-Einstein condensates.

Dynamics of nonlinear Schrödinger breathers in a potential trap.

We consider the evolution of the 2-soliton (breather) of the nonlinear Schrödinger equation on a semi-infinite line with the zero boundary condition and a linear potential, which corresponds to the gravity field in the presence of a hard floor. This setting can be implemented in atomic Bose-Einstein condensates, and in a nonlinear planar waveguide in optics. In the absence of the gravity, repulsion of the breather from the floor leads to its splitting into constituent fundamental solitons, if the initial distance from the floor is smaller than a critical value; otherwise, the moving breather persists. In the presence of gravity, the breather always splits into a pair of "co-hopping" fundamental solitons, which may be frequency locked in the form of a quasi-breather, or unlocked, forming an incoherent pseudo-breather. Some essential results are obtained in an analytical form, in addition to the systematic numerical investigation.

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 in 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.

Polarization dynamics in twisted fiber amplifiers: a non-Hermitian nonlinear dimer model.

We study continuous-wave light propagation through a twisted birefringent single-mode fiber amplifier with saturable nonlinearity. The corresponding coupled-mode system is isomorphic to a non-Hermitian nonlinear dimer and gives rise to analytic polarization-mode dynamics. It provides an optical simulation of the semi-classical non-Hermitian Bose-Hubbard model and suggests its use for the design of polarization circulators and filters, as well as sources of polarized light.

Vortex-soliton complexes in coupled nonlinear Schrödinger equations with unequal dispersion coefficients.

We consider a two-component, two-dimensional nonlinear Schrödinger system with unequal dispersion coefficients and self-defocusing nonlinearities, chiefly with equal strengths of the self- and cross-interactions. In this setting, a natural waveform with a nonvanishing background in one component is a vortex, which induces an effective potential well in the second component, via the nonlinear coupling of the two components. We show that the potential well may support not only the fundamental bound state, but also multiring excited radial state complexes for suitable ranges of values of the dispersion coefficient of the second component. We systematically explore the existence, stability, and nonlinear dynamics of these states. The complexes involving the excited radial states are weakly unstable, with a growth rate depending on the dispersion of the second component. Their evolution leads to transformation of the multiring complexes into stable vortex-bright solitons ones with the fundamental state in the second component. The excited states may be stabilized by a harmonic-oscillator trapping potential, as well as by unequal strengths of the self- and cross-repulsive nonlinearities.

Dynamics of dipoles and vortices in nonlinearly coupled three-dimensional field oscillators.

The dynamics of a pair of harmonic oscillators represented by three-dimensional fields coupled with a repulsive cubic nonlinearity is investigated through direct simulations of the respective field equations and with the help of the finite-mode Galerkin approximation (GA), which represents the two interacting fields by a superposition of 3+3 harmonic-oscillator p-wave eigenfunctions with orbital and magnetic quantum numbers l=1 and m=1, 0, -1. The system can be implemented in binary Bose-Einstein condensates, demonstrating the potential of the atomic condensates to emulate various complex modes predicted by classical field theories. First, the GA very accurately predicts a broadly degenerate set of the system's ground states in the p-wave manifold, in the form of complexes built of a dipole coaxial with another dipole or vortex, as well as complexes built of mutually orthogonal dipoles. Next, pairs of noncoaxial vortices and/or dipoles, including pairs of mutually perpendicular vortices, develop remarkably stable dynamical regimes, which feature periodic exchange of the angular momentum and periodic switching between dipoles and vortices. For a moderately strong nonlinearity, simulations of the coupled-field equations agree very well with results produced by the GA, demonstrating that the dynamics is accurately spanned by the set of six modes limited to l=1.

Rogue waves for a system of coupled derivative nonlinear Schrödinger equations.

Rogue waves (RWs) are unexpectedly strong excitations emerging from an otherwise tranquil background. The nonlinear Schrödinger equation (NLSE), a ubiquitous model with wide applications to fluid mechanics, optics, plasmas, etc., exhibits RWs only in the regime of modulation instability (MI) of the background. For a system of multiple waveguides, the governing coupled NLSEs can produce regimes of MI and RWs, even if each component has dispersion and cubic nonlinearity of opposite signs. A similar effect is demonstrated here for a system of coupled derivative NLSEs (DNLSEs) where the special feature is the nonlinear self-steepening of narrow pulses. More precisely, these additional regimes of MI and RWs for coupled DNLSEs depend on the mismatch in group velocities between the components, and the parameters for cubic nonlinearity and self-steepening. RWs considered in this paper differ from those of the NLSEs in terms of the amplification ratio and criteria of existence. Applications to optics and plasma physics are discussed.

Soliton nanoantennas in two-dimensional arrays of quantum dots.

We consider two-dimensional (2D) arrays of self-organized semiconductor quantum dots (QDs) strongly interacting with electromagnetic field in the regime of Rabi oscillations. The QD array built of two-level states is modelled by two coupled systems of discrete nonlinear Schrödinger equations. Localized modes in the form of single-peaked fundamental and vortical stationary Rabi solitons and self-trapped breathers have been found. The results for the stability, mobility and radiative properties of the Rabi modes suggest a concept of a self-assembled 2D soliton-based nano-antenna, which is stable against imperfections In particular, we discuss the implementation of such a nano-antenna in the form of surface plasmon solitons in graphene, and illustrate possibilities to control their operation by means of optical tools.

Interactions of bright and dark solitons with localized PT-symmetric potentials.

We study collisions of moving nonlinear-Schrödinger solitons with a PT-symmetric dipole embedded into the one-dimensional self-focusing or defocusing medium. Accurate analytical results are produced for bright solitons, and, in a more qualitative form, for dark ones. In the former case, an essential aspect of the approximation is that it must take into regard the intrinsic chirp of the soliton, thus going beyond the framework of the simplest quasi-particle description of the soliton's dynamics. Critical velocities separating reflection and transmission of the incident bright solitons are found by means of numerical simulations, and in the approximate semi-analytical form. An exact solution for the dark soliton pinned by the complex PT-symmetric dipole is produced too.

Dark-bright solitons in coupled nonlinear Schrödinger equations with unequal dispersion coefficients.

We study a two-component nonlinear Schrödinger system with equal, repulsive cubic interactions and different dispersion coefficients in the two components. We consider states that have a dark solitary wave in one component. Treating it as a frozen one, we explore the possibility of the formation of bright-solitonic structures in the other component. We identify bifurcation points at which such states emerge in the bright component in the linear limit and explore their continuation into the nonlinear regime. An additional analytically tractable limit is found to be that of vanishing dispersion of the bright component. We numerically identify regimes of potential stability, not only of the single-peak ground state (the dark-bright soliton), but also of excited states with one or more zero crossings in the bright component. When the states are identified as unstable, direct numerical simulations are used to investigate the outcome of the instability development. Although our principal focus is on the homogeneous setting, we also briefly touch upon the counterintuitive impact of the potential presence of a parabolic trap on the states of interest.

Radiating subdispersive fractional optical solitons.

It was recently found [Fujioka et al., Phys. Lett. A 374, 1126 (2010)] that the propagation of solitary waves can be described by a fractional extension of the nonlinear Schrödinger (NLS) equation which involves a temporal fractional derivative (TFD) of order α > 2. In the present paper, we show that there is also another fractional extension of the NLS equation which contains a TFD with α < 2, and in this case, the new equation describes the propagation of radiating solitons. We show that the emission of the radiation (when α < 2) is explained by resonances at various frequencies between the pulses and the linear modes of the system. It is found that the new fractional NLS equation can be derived from a suitable Lagrangian density, and a fractional Noether's theorem can be applied to it, thus predicting the conservation of the Hamiltonian, momentum and energy.

Bright solitons from the nonpolynomial Schrödinger equation with inhomogeneous defocusing nonlinearities.

Extending the recent work on models with spatially nonuniform nonlinearities, we study bright solitons generated by the nonpolynomial self-defocusing (SDF) nonlinearity in the framework of the one-dimensional (1D) Muñoz-Mateo-Delgado (MM-D) equation (the 1D reduction of the Gross-Pitaevskii equation with the SDF nonlinearity), with the local strength of the nonlinearity growing at |x|→∞ faster than |x|. We produce numerical solutions and analytical ones, obtained by means of the Thomas-Fermi approximation, for nodeless ground states and for excited modes with one, two, three and four nodes, in two versions of the model, with steep (exponential) and mild (algebraic) nonlinear-modulation profiles. In both cases, the ground states and the single-node ones are completely stable, while the stability of the higher-order modes depends on their norm (in the case of the algebraic modulation, they are fully unstable). Unstable states spontaneously evolve into their stable lower-order counterparts.

Trapping of light in solitonic cavities and its role in the supercontinuum generation.

We demonstrate that the fission of higher-order N-solitons with a subsequent ejection of fundamental quasi-solitons creates cavities formed by a pair of solitary waves with dispersive light trapped between them. As a result of multiple reflections of the trapped light from the bounding solitons which act as mirrors, they bend their trajectories and collide. In the spectral domain, the two solitons receive blue and red wavelength shifts, and the spectrum of the trapped light alters as well. This phenomenon strongly affects spectral characteristics of the generated supercontinuum. Consideration of the system's parameters which affect the creation of the cavity reveals possibilities of predicting and controlling soliton-soliton collisions induced by multiple reflections of the trapped light.

Soliton interaction mediated by cascaded four wave mixing with dispersive waves.

We demonstrate that trapping of dispersive waves between two optical solitons takes place when resonant scattering of the waves on the solitons leads to nearly perfect reflections. The momentum transfer from the radiation to solitons results in their mutual attraction and a subsequent collision. The spectrum of the trapped radiation can either expand or shrink in the course of the propagation, which is controlled by arranging either collision or separation of the solitons.

Symmetry breaking in linearly coupled Korteweg-de Vries systems.

We consider solitons in a system of linearly coupled Korteweg-de Vries (KdV) equations, which model two-layer settings in various physical media. We demonstrate that traveling symmetric solitons with identical components are stable at velocities lower than a certain threshold value. Above the threshold, which is found exactly, the symmetric modes are unstable against spontaneous symmetry breaking, which gives rise to stable asymmetric solitons. The shape of the asymmetric solitons is found by means of a variational approximation and in the numerical form. Simulations of the evolution of an unstable symmetric soliton sometimes produce its breakup into two different asymmetric modes. Collisions between moving stable solitons, symmetric and asymmetric ones, are studied numerically, featuring noteworthy features. In particular, collisions between asymmetric solitons with identical polarities are always elastic, while in the case of opposite polarities the collision leads to a switch of the polarities of both solitons. Three-soliton collisions are studied too, featuring quite complex interaction scenarios.

Rabi flopping induces spatial demixing dynamics.

We experimentally investigate the mixing and demixing dynamics of Bose-Einstein condensates in the presence of a linear coupling between two internal states. The observed amplitude reduction of the Rabi oscillations can be understood as a result of demixing dynamics of dressed states as experimentally confirmed by reconstructing the spatial profile of dressed state amplitudes. The observations are in quantitative agreement with numerical integration of coupled Gross-Pitaevskii equations without free parameters, which also reveals the criticality of the dynamics on the symmetry of the system. Our observations demonstrate new possibilities for changing effective atomic interactions and studying critical phenomena.

From one- to two-dimensional solitons in the Ginzburg-Landau model of lasers with frequency-selective feedback.

We use the cubic complex Ginzburg-Landau equation linearly coupled to a dissipative linear equation as a model for lasers with an external frequency-selective feedback. This system may also serve as a general pattern-formation model in media driven by an intrinsic gain and selective feedback. While, strictly speaking, the approximation of the laser nonlinearity by a cubic term is only valid for small field intensities, it qualitatively reproduces results for dissipative solitons obtained in models with a more complex nonlinearity in the whole parameter region where the solitons exist. The analysis is focused on two-dimensional stripe-shaped and vortex solitons. An analytical expression for the stripe solitons is obtained from the known one-dimensional soliton solution, and its relation with vortex solitons is highlighted. The radius of the vortices increases linearly with their topological charge m, therefore the stripe-shaped soliton may be interpreted as the vortex with m=∞, and, conversely, vortex solitons can be realized as unstable stripes bent into stable rings. The results for the vortices are applicable for a broad class of physical systems.

Chaotic solitons in the quadratic-cubic nonlinear Schrödinger equation under nonlinearity management.

We analyze the response of rational and regular (hyperbolic-secant) soliton solutions of an extended nonlinear Schrödinger equation (NLSE) which includes an additional self-defocusing quadratic term, to periodic modulations of the coefficient in front of this term. Using the variational approximation (VA) with rational and hyperbolic trial functions, we transform this NLSE into Hamiltonian dynamical systems which give rise to chaotic solutions. The presence of chaos in the variational solutions is corroborated by calculating their power spectra and the correlation dimension of the Poincaré maps. This chaotic behavior (predicted by the VA) is not observed in the direct numerical solutions of the NLSE when rational initial conditions are used. The solitary-wave solutions generated by these initial conditions gradually decay under the action of the nonlinearity management. On the contrary, the solutions of the NLSE with exponentially localized initial conditions are robust solitary-waves with oscillations consistent with a chaotic or a complex quasiperiodic behavior.