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We analyze two-component spatial optical vortex solitons supported by parametric wave mixing processes in a nonlinear bulk medium. We study two distinct cases of such localised waves, namely, parametric vortex solitons due to phase-matched second-harmonic generation in an optical medium with competing quadratic and cubic nonlinear response, and vortex solitons in the presence of third-harmonic generation in a cubic medium. We find, analytically and numerically, the structure of two-component vortex solitons, and also investigate modulational instability of their plane-wave background. In particular, we predict and analyze in detail novel types of vortex solitons, a `halo-vortex', consisting of a two-component vortex core surrounded by a bright ring of its harmonic field, and a `ring-vortex' soliton which is a vortex in a harmonic field that guides a ring-like localized mode of the fundamental-frequency field.

A model for a non-Kerr cylindrical nematic fiber is presented. We use the multiple scales method to show the possibility of constructing different kinds of wavepackets of transverse magnetic (TM) modes propagating through the fiber. This procedure allows us to generate different hierarchies of nonlinear partial differential equations (PDEs) which describe the propagation of optical pulses along the fiber. We go beyond the usual weakly nonlinear limit of a Kerr medium and derive an extended Nonlinear Schrodinger equation (eNLS) with a third order derivative nonlinearity, governing the dynamics for the amplitude of the wavepacket. In this derivation the dispersion, self-focussing and diffraction in the nematic are taken into account. Although the resulting nonlinear $PDE$ may be reduced to the modified Korteweg de Vries equation (mKdV), it also has additional complex solutions which include two-parameter families of bright and dark complex solitons. We show analytically that under certain conditions, the bright solitons are actually double embedded solitons. We explain why these solitons do not radiate at all, even though their wavenumbers are contained in the linear spectrum of the system. Finally, we close the paper by making comments on the advantages as well as the limitations of our approach, and on further generalizations of the model and method presented.

In previous work we showed that bright atomic solitons can arise in spinor Bose-Einstein condensates in the form of gap solitons even for repulsive many-body interactions. Here we further explore the properties of atomic gap solitons and show that their internal structure can be used to both excite them and control their center-of-mass motion using applied laser and magnetic fields. As an illustration we demonstrate a nonlinear atom-optical Mach-Zehnder interferometer based on gap solitons.

We investigate effects of the application of a kick to one-dimensional matter-wave solitons in a self-attractive Bose-Einstein condensate trapped in a optical lattice. The resulting soliton's dynamics is studied within the framework of the time-dependent nonpolynomial Schrodinger equation. The crossover from the pinning to quasi-free motion crucially depends on the size of the kick, strength of the self-attraction, and parameters of the optical lattice.

We study interactions of extremely short three-dimensional bipolar electromagnetic pulses propagating towards each other in an array of semiconductor carbon nanotubes, along any direction perpendicular to their axes. The analysis provides a full account of the effects of the nonuniformity of the pulses' fields along the axes. The evolution of the electromagnetic field and charge density in the sample is derived from the Maxwell's equations and the continuity equation, respectively. In particular, we focus on indirect interaction of the pulses via the action of their fields on the electronic subsystem of the nanotube array. Changes in the shape of pulses in the course of their propagation and interaction are analyzed by calculating and visualizing the distribution of the electric field in the system. The numerical analysis reveals a possibility of stable post-collision propagation of pulses over distances much greater than their sizes.

Continuous-wave laser driven Kerr-nonlinear, optical microresonators have enabled a variety of novel applications and phenomena including the generation of optical frequency combs, ultra-low noise microwaves, as well as, ultra-short optical pulses. In this work we break with the paradigm of the continuous-wave optical drive and use instead periodic, pico-second optical pulses. We observe the deterministic generation of stable femtosecond dissipative cavity solitons on-top of the resonantly enhanced driving pulse. Surprisingly, the soliton pulse locks to the driving pulse enabling direct all-optical control of both the soliton's repetition rate and carrier-envelope offset frequency without the need for any actuation on the microresonator. When compared to both continuous-wave driven microresonators and non-resonant pulsed supercontinuum generation, this new approach is substantially more efficient and can yield broadband frequency combs at femto-Joule driving pulse energies and average laser powers significantly below the parametric threshold power of continuous-wave driven microresonators. The presented results bridge the fields of continuous-wave driven resonant and pulse-driven non-resonant nonlinear optics. They enables micro-photonic pulse compression, ultra-efficient low noise frequency comb and resonant supercontinuum generation for applications including optical data transfer and optical spectroscopy. From a scientific perspective the results open a new horizon for nonlinear photonics driven by temporally and spectrally structured light.

The existence and stability of solitons in Bose-Einstein condensates with attractive inter-atomic interactions, described by the Gross-Pitaevskii equation with a three-dimensional (3D) periodic potential, are investigated in a systematic form. We find a one-parameter family of stable 3D solitons in a certain interval of values of their norm, provided that the strength of the potential exceeds a threshold value. The minimum number of $^{7}$Li atoms in the stable solitons is 60, and the energy of the soliton at the stability threshold is $\approx 6$ recoil energies in the lattice. The respective energy-vs.-norm diagram features two cuspidal points, resulting in a typical \textit{swallowtail pattern}, which is a generic feature of 3D solitons supported by low- (2D) or fully-dimensional lattice potentials.

A model of a Bose-Einstein condensate in a ring optical lattice with atomic dissipations applied at a stationary or at a moving location on the ring is presented. The localized dissipation is shown to generate and stabilize both stationary and traveling lattice solitons. Among many localized solutions, we have generated spatially stationary quasiperiodic lattice soltions and a family of traveling lattice solitons with two intensity peaks per potential well with no counterpart in the discrete case. Collisions between traveling and stationary lattice solitons as well as between two traveling lattice solitons display a critical dependence from the lattice depth. Stable counterpropagating solitons in ring lattices can find applications in gyroscope interferometers with ultra-cold gases.

By solving the three-dimensional Gross-Pitaevskii equation we generate two-dimensional axially-symmetric and anisotropic dipolar Bose-Einstein condensate bright solitons, for repulsive atomic interaction, stabilized by only a weak one-dimensional optical lattice (OL) aligned along and perpendicular, respectively, to the dipole polarization direction. In the former case vortex solitons can also be created. We show that it is possible to make a stable array of small interacting axially-symmetric dipolar solitons put on alternate OL sites. Further, we demonstrate the elastic nature of the collision of two such solitons.

We study numerically and experimentally supercontinuum generation in optical fibers with dark and bright solitons simultaneously contributing into the spectral broadening and dispersive wave generation. We report a novel type of weak trapped radiation arising due to interaction of bright solitons with the dark soliton background. This radiation expresses itself as two pulses with the continuously shifting spectra constituting the short and long wavelength limits of the continuum. Our theoretical and experimental results are in good agreement.

Using a tight-binding Hamiltonian we study the Peierls distorted form of polyaniline, the pernigraniline base (PNB) polymer. Both bond order and ring torsional degrees of freedom are considered in the conformational relaxation. The geometrical and electronic structures of the localized solitonic and polaronic excitations are calculated. Comparison with available optical and magnetic measurements is made. Particular emphasis is placed on the spectroscopic and magnetic features of neutral solitons.

We analyze the existence and stability of bright, dark, and gap matter-wave solitons in optical superlattices. Then, using these properties, we show that (time-dependent) ``dynamical superlattices'' can be used to controllably place, guide, and manipulate these solitons. In particular, we use numerical experiments to displace solitons by turning on a secondary lattice structure, transfer solitons from one location to another by shifting one superlattice substructure relative to the other, and implement solitonic ``path-following'', in which a matter wave follows the time-dependent lattice substructure into oscillatory motion.

We study propagation of light in nonlinear diffraction-managed photonic lattices created with arrays of periodically-curved coupled optical waveguides which were fabricated using femtosecond laser writing in silica glass, and titanium indiffusion in LiNbO3 crystals. We identify different regimes of the nonlinear propagation of light beams depending on the input power, and present the first experimental observation of diffraction-managed solitons, which are formed as a result of the interplay between the engineered beam diffraction and nonlinear self-focusing or defocusing. We observe that in self-collimating structures where linear diffraction is suppressed, a novel regime of nonlinear beam diffusion takes place at the intermediate powers before the lattice soliton is formed at higher powers.

The effects caused by nonresonant nonlinear interaction between noncollinear self-focusing beams are considered in 2D optical samples using multi-scale analysis. The analytical expression for the beams trajectories shift due to the mutual interaction is derived, and the range of parameters is given beyond which the mentioned consideration fails. We compare our results with the naive geometrical optics model. It is shown that these two approaches give the same results. This justifies the use of geometrical optics approach for description of elastic and almost-elastic collision processes both in Kerr and saturable nonlinear media.

We consider a dynamical model for a Fermi gas in the Bardeen-Cooper-Schrieffer (BCS) superfluid state, trapped in a combination of a 1D or 2D optical lattice (OL) and a tight parabolic potential acting in the transverse direction(s). The model is based on an equation for the order parameter (wave function), which is derived from the energy density for the weakly coupled BCS superfluid. The equation includes a nonlinear self-repulsive term of power 7/3, which accounts for the Fermi pressure. Reducing the equation to the 1D or 2D form, we construct families of stable 1D and 2D gap solitons (GSs) by means of numerical simulations, which are guided by the variational approximation (VA). The GSs are, chiefly, compact objects trapped in a single cell of the OL potential. In the linear limit, the VA predicts almost exact positions of narrow Bloch bands that separate the semi-infinite and first finite gaps, as well as the first and second finite ones. Families of stable even and odd bound states of 1D GSs are constructed too. We also demonstrate that the GS can be dragged without much distortion by an OL moving at a moderate velocity ($\sim $ 1 mm/s, in physical units). The predicted GSs contain $\sim 10^{3}-10^{4}$ and $\sim 10^{3}$ atoms per 1D and 2D settings, respectively.

We investigate the existence and stability of three-dimensional (3D) solitons supported by cylindrical Bessel lattices (BLs) in self-focusing media. If the lattice strength exceeds a threshold value, we show numerically, and using the variational approximation, that the solitons are stable within one or two intervals of values of their norm. In the latter case, the Hamiltonian-vs.-norm diagram has a "swallowtail" shape, with three cuspidal points. The model applies to Bose-Einstein condensates (BECs) and to optical media with saturable nonlinearity, suggesting new ways of making stable 3D BEC solitons and "light bullets" of an arbitrary size.

We study the gap solitons and nonlinear Bloch waves of interacting bosons in one-dimensional optical lattices, taking into account the interaction from the weak to the strong limits. It is shown that composition relation between the gap solitons and nonlinear Bloch waves exists for the whole span of the interaction strength. The linear stability analysis indicates that the gap solitons are stable when their energies are near the bottom of the linear Bloch band gap. By increasing the interaction strength, the stable gap solitons can turn into unstable. It is argued that the stable gap solitons can easily be formed in a weakly interacting system with energies near the bottoms of the lower-level linear Bloch band gaps.

We show that gap-acoustic solitons, i.e., optical gap solitons with electrostrictive coupling to sound modes, can be produced with velocities down to less than 2.5% of the speed of light using a fiber Bragg grating that is linearly coupled to a non-Bragg fiber over a finite domain. Forward- and backward-moving light pulses in the non-Bragg fiber that reach the coupling region simultaneously couple into the Bragg fiber and form a moving soliton, which then propagates beyond the coupling region.

We present a fully quantum many-body treatment of dark solitons formed by ultracold bosonic atoms in one-dimensional optical lattices. Using time-evolving block decimation to simulate the single-band Bose-Hubbard Hamiltonian, we consider the quantum dynamics of density and phase engineered dark solitons as well as the quantum evolution of mean-field dark solitons injected into the quantum model. The former approach directly models how one may create quantum entangled dark solitons in experiment. While we have already presented results regarding the latter approach elsewhere [Phys. Rev. Lett. {\bf 103}, 140403 (2009)], we expand upon those results in this work. In both cases, quantum fluctuations cause the dark soliton to fill in and may induce an inelasticity in soliton-soliton collisions. Comparisons are made to the Bogoliubov theory which predicts depletion into an anomalous mode that fills in the soliton. Our many-body treatment allows us to go beyond the Bogoliubov approximation and calculate explicitly the dynamics of the system's natural orbitals.

Some selected important properties of photorefractive spatial solitons and their applications have been reviewed in the present paper. Using band transport model, the governing principle of photorefractive nonlinearity has been addressed and nonlinear dynamical equations of spatial solitons owing to this nonlinearity have been discussed. Mechanisms of formation of screening and photovoltaic solitons of three different configurations i.e., bright, dark and grey varieties have been examined. Incoherently coupled vector solitons due to single and two-photon photorefractive phenomena have been highlighted. Modulation instability, which is precursor to soliton formation has been also discused briefly. Finally possible applications of photorefractive spatial solitons have been highlighted.

A numerical and analytical study of soliton propagation and generation in a Raman amplifier is performed. Parameters are amplitude and temporal profile of input optical fields, Raman line width and detuning. Soliton excitations will develop, if a phase shift of nearly 180 degrees is introduced into the Stokes seed beam. In the absence of detuning and for an instantaneous phase shift of exactly 180 degrees the solitons wil be stable. In the presence of detuning and for non instantaneous phase shifts of less than 180 degrees the solitons will be unstable and decay for sufficiently large gain. Limiting values are about 10% detuning relative to the Raman line width, and a fractional, unshifted Stokes intensity of about 1%. Stable solitons will show temporal narrowing with a width inversely proportional to the square root of the total gain. For large gain the final width is independent of the initial width. Unstable solitons will reach minimal width at a relative amplitude of .5, and begin to broaden on further propagation. The minimal width is proportional to the initial width and the initial relative amplitude. Keywords: Raman Scattering; Solitions; Ultrashort Optical Pulses.

We develop one numerical method to compute black and gray solitons with Kerr-type nonlocal nonlinearity. As two examples of nonlocal cases, the gray soliton with exponentially decaying nonlocal response or with Gaussian nonlocal response are discussed. For such two nonlocal cases, the analytical form of the tails of nonlocal gray soliton is presented and the analytical relationship for the maximal transverse velocity of nonlocal gray soliton to the characteristic nonlocal length is obtained.

We prove that spatial Kerr solitons, usually obtained in the frame of nonlinear Schroedinger equation valid in the paraxial approximation, can be found in a generalized form as exact solutions of Maxwell's equations. In particular, they are shown to exist, both in the bright and dark version, as linearly polarized exactly integrable one-dimensional solitons, and to reduce to the standard paraxial form in the limit of small intensities. In the two-dimensional case, they are shown to exist as azimuthally polarized circularly symmetric dark solitons. Both one and two-dimensional dark solitons exhibit a characteristic signature in that their asymptotic intensity cannot exceed a threshold value in correspondence of which their width reaches a minimum subwavelength value.

We propose a way to control solitons in $\chi ^{(2)}$ (quadratically-nonlinear) systems by means of periodic modulation imposed on the phase-mismatch parameter ("mismatch management", MM). It may be realized in the co-transmission of fundamental-frequency (FF) and second-harmonic (SH) waves in a planar optical waveguide via a long-period modulation of the usual quasi-phase-matching pattern of ferroelectric domains. The MM may also be implemented by dint of the Feshbach resonance in a harmonically-modulated magnetic field in a hybrid atomic-molecular Bose-Einstein condensate (BEC), with the atomic and molecular mean fields (MFs) playing the roles of the FF and SH, respectively. The problem is analyzed by two methods. First, we identify stability regions for spatial solitons in the MM system, in terms of the MM amplitude and period, using the MF equations for spatially-inhomogeneous configurations. In particular, an instability enclave is found inside the stability area.The robustness of the solitons is also tested against variation of the shape of the input pulse, and a threshold for the formation of stable solitons is found in terms of its power. Interactions between solitons are virtually unaffected by the MM. The second method (\textit{parametric approximation}), going beyond the MF description, is developed for spatially-homogeneous states. It demonstrates that the MF description is valid for large modulation periods, while at smaller periods the non-MF component acquires gain, which implies destruction of MF under the action of the high-frequency MM.

We demonstrate numerically that partially incoherent light can be trapped in the spectral band gaps of a photonic lattice, creating partially incoherent multi-component spatial optical solitons in a self-defocusing nonlinear periodic medium. We find numerically such incoherent multi-gap optical solitons and discuss how to generate them in experiment by interfering incoherent light beams at the input of a nonlinear periodic medium.

We study the properties of linear and nonlinear waveguides induced by optical vortex solitons in both self-defocusing and self-focusing nonlinear media, for the case of a saturable nonlinear response. We demonstrate that the vortex-induced waveguides can guide both fundamental and first-order guided modes which together with the vortex may form, for larger amplitudes of the guide modes, different types of composite vortex-mode vector solitons. In the case of the self-focusing saturable medium, we demonstrate that a large-amplitude guided mode can stabilize the ring-like vortex structure which usually decays due to azimuthal modulational instability.

We introduce a novel class of parametric optical solitons supported simultaneously by two second-order nonlinear cascading processes, second-harmonic generation and sum-frequency mixing. We obtain, analytically and numerically, the solutions for three-wave spatial solitons and show that the presence of an additional cascading mechanism can change dramatically the properties and stability of two-wave quadratic solitary waves.

We introduce a novel class of parametric optical solitons supported simultaneously by two second-order nonlinear cascading processes, second-harmonic generation and sum-frequency mixing. We obtain, analytically and numerically, the solutions for three-wave spatial solitons and show that the presence of an additional cascading mechanism can change dramatically the properties and stability of two-wave quadratic solitary waves.

Inverted nonlinear photonic crystals are the crystals featuring competition between linear and nonlinear lattices, with minima of the linear potential coinciding with maxima of the nonlinear pseudopotential, and vice versa. Traditional inverted nonlinear photonic crystals only have two channels, and can be attained experimentally by means of Rhodamine B (RhB, a dye featuring saturable absorption) doped into the SU-8 polymer. In this paper, a new type of inverted nonlinear photonic crystal is constructed by juxtaposing three kinds of channels into a period. These three channels are a purely linear channel, a saturable self-focusing nonlinear channel, and a saturable self-defocusing nonlinear channel. This optical device is assumed to be fabricated by means of SU-8 polymer material periodically doped with two types of active dyes. The nonlinear propagation of a light field inside this device (passing along the channel) can be described by a nonlinear Schrodinger equation. Stable multi-peak fundamental and dipole solitons are found in the first gap of the system. These solitons sufficiently exhibit some interesting digital properties, which may have potential in optical communications.

We suggest and study the stable disk- and cigar-shaped gap solitons of a dipolar Bose-Einstein condensate of $^{52}$Cr atoms localized in the lowest band gap by three optical-lattice (OL) potentials along orthogonal directions. The one-dimensional version of these solitons of experimental interest confined by an OL along the dipole moment direction and harmonic traps in transverse directions is also considered. Important dynamics of (i) breathing oscillation of a gap soliton upon perturbation and (ii) dragging of a gap soliton by a moving lattice along axial $z$ direction demonstrates the stability of gap solitons. A movie clip of dragging of three-dimensional gap soliton is included.

We present the results of asymptotic and numerical analysis of dissipative Kerr solitons in whispering gallery mode microresonators influenced by higher order dispersive terms leading to the appearance of a dispersive wave (Cherenkov radiation). Combining direct perturbation method with the method of moments we find expressions for the frequency, strength, spectral width of the dispersive wave and soliton velocity. Mutual influence of the soliton and dispersive wave was studied. The formation of the dispersive wave leads to a shift of the soliton spectrum maximum from the pump frequency (spectral recoil), while the soliton displaces the dispersive wave spectral peak from the zero dispersion point.

Dissipative solitons are self-localized structures resulting from a double balance between dispersion and nonlinearity as well as dissipation and a driving force. They occur in a wide variety of fields ranging from optics, hydrodynamics to chemistry and biology. Recently, significant interest has focused on their temporal realization in driven optical microresonators, known as dissipative Kerr solitons. They provide access to coherent, chip-scale optical frequency combs, which have already been employed in optical metrology, data communication and spectroscopy. Such Kerr resonator systems can exhibit numerous localized intracavity patterns and provide rich insights into nonlinear dynamics. A particular class of solutions consists of breathing dissipative solitons, representing pulses with oscillating amplitude and duration, for which no comprehensive understanding has been presented to date. Here, we observe and study single and multiple breathing dissipative solitons in two different microresonator platforms: crystalline $\mathrm{MgF_2}$ resonator and $\mathrm{Si_3N_4}$ integrated microring. We report a deterministic route to access the breathing state, which allowed for a detailed exploration of the breathing dynamics. In particular, we establish the link between the breathing frequency and two system control parameters - effective pump laser detuning and pump power. Using a fast detection, we present a direct observation of the spatiotemporal dynamics of individual solitons, revealing irregular oscillations and switching. An understanding of breathing solitons is not only of fundamental interest concerning nonlinear systems close to critical transition, but also relevant for applications to prevent breather-induced instabilities in soliton-based frequency combs.

We show the formation of bright and dark slow optical solitons based on intersubband transitions in a semiconductor quantum well (SQW). Using the coupled Schrodinger-Maxwell approach, we provide both analytical and numerical results. Such a nonlinear optical process may be used for the control technology of optical delay lines and optical buffers in the SQW solid-state system. With appropriate parameters, we also show the generation of a large cross-phase modulation (XPM). Since the the intersubband energy level can be easily tuned by an external bias voltage, the present investigation may open the possibility for electrically controlled phase modulator in the solid-state system.

We present a theory describing trapping of the normally dispersive radiation by the Raman solitons in optical fibers. Frequency of the radiation component is continuously blue shifting, while the soliton is red shifting. Underlying physics of the trapping effect is in the existence of the inertial gravity-like force acting on light in the accelerating frame of reference. We present analytical calculations of the rate of the opposing frequency shifts of the soliton and trapped radiation and find it to be greater than the rate of the red shift of the bare Raman soliton. Our findings are essential for understanding of the continuous shift of the high frequency edge of the supercontinuum spectra generated in photonic crystal fibers towards higher frequencies.

We study inelastic collisions of counter-propagating self-induced transparency solitons in a homogeneously broadened two-level medium. The energy of the pulse can be almost totally absorbed in the medium due to asymmetric collision with a properly chosen control pulse. The medium state thus prepared demonstrates the property of an all-optical diode which transmits pulses from one direction and blocks from another. The saturation process of a controlled absorption effect, local-field correction influence, and the parameter ranges for the diode action are studied as well.

Pulse broadening for optical solitons due to birefringence is investigated. We present an analytical solution which describes the propagation of solitons in birefringent optical fibers. The special solutions consist of a combination of purely solitonic terms propagating along the principal birefringence axes and soliton-soliton interaction terms. The solitonic part of the solutions indicates that the decay of initially localized pulses could be due to different propagation velocities along the birefringence axes. We show that the disintegration of solitonic pulses in birefringent optical fibers can be caused by two effects. The first effect is similar as in linear birefringence and is related to the unequal propagation velocities of the modes along the birefringence axes. The second effect is related to the nonlinear soliton-soliton interaction between the modes, which makes the solitonic pulse-shape blurred.

We have reviewed recent developments of some aspects of optical spatial solitons in photorefractive media. Underlying principles governing the dynamics of photorefractive nonlinearity have been discussed using band transport model. Nonlinear dynamical equations for propagating solitons have been derived considering single as well as two-photon photorefractive processes. Fundamental properties of three types of solitons, particularly, screening, photovoltaic and screening photovoltaic solitons have been considered. For each type of solitons, three different configurations i.e., bright, dark and gray varieties have been considered. Mechanisms of formation of these solitons due to single as well as two-photon photorefractive processes have been considered and their self bending discussed. Vector solitons, particularly, incoherently coupled solitons due to single photon and two-photon photorefractive phenomena have been highlighted. Existence of some missing solitons have been also pointed out.

We study the dynamics of one-dimensional solitons in the attractive and repulsive Bose-Einstein condensates (BECs) loaded into an optical lattice (OL), which is combined with an external parabolic potential. First, we demonstrate analytically that, in the repulsive BEC, where the soliton is of the gap type, its effective mass is \emph{negative}. This gives rise to a prediction for the experiment: such a soliton cannot be not held by the usual parabolic trap, but it can be captured (performing harmonic oscillations) by an anti-trapping inverted parabolic potential. We also study the motion of the soliton a in long system, concluding that, in the cases of both the positive and negative mass, it moves freely, provided that its amplitude is below a certain critical value; above it, the soliton's velocity decreases due to the interaction with the OL. At a late stage, the damped motion becomes chaotic. We also investigate the evolution of a two-soliton pulse in the attractive model. The pulse generates a persistent breather, if its amplitude is not too large; otherwise, fusion into a single fundamental soliton takes place. Collisions between two solitons captured in the parabolic trap or anti-trap are considered too. Depending on their amplitudes and phase difference, the solitons either perform stable oscillations, colliding indefinitely many times, or merge into a single soliton. Effects reported in this work for BECs can also be formulated for optical solitons in nonlinear photonic crystals. In particular, the capture of the negative-mass soliton in the anti-trap implies that a bright optical soliton in a self-defocusing medium with a periodic structure of the refractive index may be stable in an anti-waveguide.

The experimental demonstration of Bose-Einstein condensation in atomic vapors has rapidly lead to new advances in atom optics. It is now well established that two-body collisions play for matter waves a role analogous to that of a Kerr nonlinear crystal in optics. In particular, it is known that the nonlinear Schr\"odinger equation which describes the condensate supports soliton solutions. For the case of repulsive interactions normally encountered in BEC experiments, the simplest solutions are dark solitons, that is, 'dips' in the density profile of the condensate. However, it is desirable to achieve the dispersionless transport of a spatially localized ensemble of atoms, rather than a 'hole'. In that case, bright solitons are much more interesting although the problem is that large condensates are necessarily associated with repulsive interactions, for which bright solitons might seem impossible. While this is true for atoms in free space, this is however not the case for atoms in suitable potentials, eg. optical lattices. This result is known from nonlinear optics, where such soliton solutions, called gap solitons, have been predicted and demonstrated. We will briefly review the basic formalism for nonlinear atom optics and summarize various soliton solutions and then describe a specific model for gap solitons in a spinor Bose-Einstein condensate.

The experimental demonstration of Bose-Einstein condensation in atomic vapors has rapidly lead to new advances in atom optics. It is now well established that two-body collisions play for matter waves a role analogous to that of a Kerr nonlinear crystal in optics. In particular, it is known that the nonlinear Schr\"odinger equation which describes the condensate supports soliton solutions. For the case of repulsive interactions normally encountered in BEC experiments, the simplest solutions are dark solitons, that is, 'dips' in the density profile of the condensate. However, it is desirable to achieve the dispersionless transport of a spatially localized ensemble of atoms, rather than a 'hole'. In that case, bright solitons are much more interesting although the problem is that large condensates are necessarily associated with repulsive interactions, for which bright solitons might seem impossible. While this is true for atoms in free space, this is however not the case for atoms in suitable potentials, eg. optical lattices. This result is known from nonlinear optics, where such soliton solutions, called gap solitons, have been predicted and demonstrated. We will briefly review the basic formalism for nonlinear atom optics and summarize various soliton solutions and then describe a specific model for gap solitons in a spinor Bose-Einstein condensate.

We study the soliton mobility in nonlocal nonlinear media with an imprinted fading optical lattice. The results show that the transverse mobility of solitons varies with the lattice decay rate and the nonlocality degree, which provides an opportunity for all-optical control of light.

We propose an experimentally relevant scheme to create stable solitons in a three-dimensional Bose-Einstein condensate confined by a one-dimensional optical lattice, using temporal modulation of the scattering length (through ac magnetic field tuned close to the Feshbach resonance). Another physical interpretation is a possibility to create stable 3D "light bullets" in an optical medium with a longitudinal alternating self-focusing/defocusing structure, and periodic modulation of the refractive index in a transverse direction. We develop a variational approximation to identify a stability region in the parametric space, and verify the existence of stable breathing solitons in direct simulations. Both methods reveal that stable solitons may be supported if the average value of the nonlinear coefficient (whose sign corresponds to attraction between atoms) and the lattice's strength exceed well-defined minimum values. Stable localized patterns may feature a multi-cell structure.

We present, experimentally and numerically, the observation of Fermi-Pasta-Ulam recurrence induced by breather solitons in a high-Q SiN microresonator. Breather solitons can be excited by increasing the pump power at a relatively small pump phase detuning in microresonators. Out of phase power evolution is observed for groups of comb lines around the center of the spectrum compared to groups of lines in the spectral wings. The evolution of the power spectrum is not symmetric with respect to the spectrum center. Numerical simulations based on the generalized Lugiato-Lefever equation are in good agreement with the experimental results and unveil the role of stimulated Raman scattering in the symmetry breaking of the power spectrum evolution. Our results shows that optical microresonators can be exploited as a powerful platform for the exploration of soliton dynamics.

Stability of solitons in parity-time (PT)-symmetric periodic potentials (optical lattices) is analyzed in both one- and two-dimensional systems. First we show analytically that when the strength of the gain-loss component in the PT lattice rises above a certain threshold (phase-transition point), an infinite number of linear Bloch bands turn complex simultaneously. Second, we show that while stable families of solitons can exist in PT lattices, increasing the gain-loss component has an overall destabilizing effect on soliton propagation. Specifically, when the gain-loss component increases, the parameter range of stable solitons shrinks as new regions of instability appear. Thirdly, we investigate the nonlinear evolution of unstable PT solitons under perturbations, and show that the energy of perturbed solitons can grow unbounded even though the PT lattice is below the phase transition point.

Recently, Biswas and Milovic [Appl. Math. Comput. 208 (2009) 209-302] have found optical one-soliton solutions of a fourth order dispersive cubic-quintic nonlinear Schr\"odinger equation. In this comment, we first show there are mistakes in the paper and demonstrate that the obtained solutions do not satisfy the considered equation. And then we reconstruct a series of analytical exact solutions by means of a direct ansatz method and F-expansion method. These solutions include solitary wave solutions of the bell shape, solitary wave solutions of the kink shape, and periodic wave solutions of Jacobian elliptic function.

Criteria for experimental observation of multi-dimensional optical solitons in media with saturable refractive nonlinearities are developed. The criteria are applied to actual material parameters (characterizing the cubic self-focusing and quintic self-defocusing nonlinearities, two-photon loss, and optical-damage threshold) for various glasses. This way, we identify operation windows for soliton formation in these glasses. It is found that two-photon absorption sets stringent limits on the windows. We conclude that, while a well-defined window of parameters exists for two-dimensional solitons (spatial or spatiotemporal), for their three-dimensional spatiotemporal counterparts such a window \emph{does not} exist, due to the nonlinear loss in glasses.

Motivated by recent experimental achievement in the work with Bose-Einstein condensates (BECs), we consider bright matter-wave solitons, in the presence of a parabolic magnetic trap and a spatially periodic optical lattice (OL), in the attractive BEC. We examine pinned states of the soliton and their stability by means of perturbation theory. The analytical predictions are found to be in good agreement with numerical simulations. We then explore possibilities to use a time-modulated OL as a means of stopping and trapping a moving soliton, and of transferring an initially stationary soliton to a prescribed position by a moving OL. We also study the emission of radiation from the soliton moving across the combined magnetic trap and OL. We find that the soliton moves freely (without radiation) across a weak lattice, but suffers strong loss for stronger OLs.

Temporal-dissipative Kerr solitons are self-localized light pulses sustained in driven nonlinear optical resonators. Their realization in microresonators has enabled compact sources of coherent optical frequency combs as well as the study of dissipative solitons. A key parameter of their dynamics is the effective-detuning of the pump laser to the thermally- and Kerr-shifted cavity resonance. Together with the free spectral range and dispersion, it governs the soliton-pulse duration, as predicted by an approximate analytical solution of the Lugiato-Lefever equation. Yet, a precise experimental verification of this relation was lacking so far. Here, by measuring and controlling the effective-detuning, we establish a new way of stabilizing solitons in microresonators and demonstrate that the measured relation linking soliton width and detuning deviates by less than 1 % from the approximate expression, validating its excellent predictive power. Furthermore, a detuning-dependent enhancement of specific comb lines is revealed, due to linear couplings between mode-families. They cause deviations from the predicted comb power evolution, and induce a detuning-dependent soliton recoil that modifies the pulse repetition-rate, explaining its unexpected dependence on laser-detuning. Finally, we observe that detuning-dependent mode-crossings can destabilize the soliton, leading to an unpredicted soliton breathing regime (oscillations of the pulse) that occurs in a normally-stable regime. Our results test the approximate analytical solutions with an unprecedented degree of accuracy and provide new insights into dissipative-soliton dynamics.

Temporal-dissipative Kerr solitons are self-localized light pulses sustained in driven nonlinear optical resonators. Their realization in microresonators has enabled compact sources of coherent optical frequency combs as well as the study of dissipative solitons. A key parameter of their dynamics is the effective-detuning of the pump laser to the thermally- and Kerr-shifted cavity resonance. Together with the free spectral range and dispersion, it governs the soliton-pulse duration, as predicted by an approximate analytical solution of the Lugiato-Lefever equation. Yet, a precise experimental verification of this relation was lacking so far. Here, by measuring and controlling the effective-detuning, we establish a new way of stabilizing solitons in microresonators and demonstrate that the measured relation linking soliton width and detuning deviates by less than 1 % from the approximate expression, validating its excellent predictive power. Furthermore, a detuning-dependent enhancement of specific comb lines is revealed, due to linear couplings between mode-families. They cause deviations from the predicted comb power evolution, and induce a detuning-dependent soliton recoil that modifies the pulse repetition-rate, explaining its unexpected dependence on laser-detuning. Finally, we observe that detuning-dependent mode-crossings can destabilize the soliton, leading to an unpredicted soliton breathing regime (oscillations of the pulse) that occurs in a normally-stable regime. Our results test the approximate analytical solutions with an unprecedented degree of accuracy and provide new insights into dissipative-soliton dynamics.

We show that the polarization domains generated in the linear optical spin-Hall effect by the analog of spin-orbit interaction for exciton-polaritons are associated with the formation of a Skyrmion lattice. In the non-linear regime, the spin anisotropy of the polariton-polariton interactions results in a spatial compression of the spin domains and in an abrupt transformation of the Skyrmions into half-solitons, associated with both the focalization of the spin currents and the emergence of a strongly anisotropic emission pattern.