Exact solutions for bright and dark solitons in spatially inhomogeneous nonlinearity

We present exact analytical results for bright and dark solitons in a type of one-dimensional spatially inhomogeneous nonlinearity. We show that the competition between a homogeneous self-defocusing (SDF) nonlinearity and a localized self-focusing (SF) nonlinearity supports stable fundamental bright solitons. For a specific choice of the nonlinear parameters, exact analytical solutions for fundamental bright solitons have been obtained. By applying both variational approximation and Vakhitov-Kolokolov stability criterion, it is found that exact fundamental bright solitons are stable. Our analytical results are also confirmed numerically. Additionally, we show that a homogeneous SF nonlinearity modulated by a localized SF nonlinearity allows the existence of exact dark solitons, for certain special cases of nonlinear parameters. By making use of linear stability analysis and direct numerical simulation, it is found that these exact dark solitons are linearly unstable.     

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.     

Stable bright and vortex solitons in photonic crystal fibers with inhomogeneous defocusing nonlinearity

We predict that a photonic crystal fiber whose strands are filled with a defocusing nonlinear medium can support stable bright solitons and also vortex solitons if the strength of the defocusing nonlinearity grows toward the periphery of the fiber. The domains of soliton existence depend on the transverse growth rate of the filling nonlinearity and nonlinearity of the core. Remarkably, solitons exist even when the core material is linear.     

Surface lattice solitons in diffusive nonlinear media

We address the properties of surface solitons supported by optical lattices imprinted in photorefractive media with asymmetrical diffusion nonlinearity. Such solitons exist only in finite gaps of the lattice spectrum. In contrast to latticeless geometries, where surface waves exist only when nonlinearity deflects light toward the material surface, the surface lattice solitons exist in settings where diffusion would cause beam bending against the surface.     

Three-dimensional vortex solitons in self-defocusing media

We demonstrate that families of vortex solitons are possible in a bidispersive three-dimensional nonlinear Schr?dinger equation. These solutions can be considered as extensions of two-dimensional dark vortex solitons which, along the third dimension, remain localized due to the interplay between dispersion and nonlinearity. Such vortex solitons can be observed in optical media with normal dispersion, normal diffraction, and defocusing nonlinearity.     

Counterpropagating spatial solitons in reflection gratings with a longitudinally modulated Kerr nonlinearity

We investigate quasi-Bragg-matched counterpropagating spatial solitons in a reflection grating in the presence of a longitudinally modulated Kerr nonlinearity. The physical interplay of linear reflection and Kerr self-focusing with the modulation in the nonlinearity yields a variety of elaborate self-action mechanisms. We first analytically predict the existence of symmetric soliton pairs supported by a pure Kerr-like effective nonlinearity. We then analytically derive two families of solitons, associated with the linear grating eigenmodes, supported by an effective "incoherent" Kerr-like coupling arising from the exact balance between the modulation in the nonlinearity and the Kerr interaction due to beam interference.     

Gap soliton in a waveguide array with a saturating defocusing nonlinearity

We investigate radiation of the solitary waves in the first band gap of the waveguide array with a defocusing nonlinearities of different types (Kerr nonlinearity and saturating nonlinearity). We confirm recent findings that gap solitons (GSs) are unstable for their eigenfrequencies around the middle of the band gap for Kerr nonlinearity. The instability is mediated by four-wave mixing process and appears in the form of radiation of solitons into mode continua of the upper and lower bands. We find that this soliton radiation is reduced (and even suppressed completely) in case of a saturating nonlinearity, resulting in substantial stabilization of the GSs.     

Observation of elliptic incoherent spatial solitons

We present the first experimental observation of spatially incoherent elliptic solitons. We use partially spatially incoherent light with anisotropic correlation statistics and observe elliptic solitons supported by the photorefractive screening nonlinearity.     

Route to nonlocality and observation of accessible solitons

We develop a general theory of spatial solitons in a liquid crystalline medium exhibiting a nonlinearity with an arbitrary degree of effective nonlocality. The model accounts the observability of accessible solitons and establishes an important link with parametric solitons.     

Attraction of nonlocal dark optical solitons

We study the formation and interaction of spatial dark optical solitons in materials with a nonlocal nonlinear response. We show that unlike in local materials, where dark solitons typically repel, the nonlocal nonlinearity leads to a long-range attraction and formation of stable bound states of dark solitons.     

Analytical solution for photorefractive screening solitons

We study formation and interaction of one-dimensional screening solitons in a photorefractive medium with sublinear dependence of the photoconductivity on light intensity. We find an exact analytical solution to the corresponding nonlinear Schrodinger equation. We show that these solitons are stable in propagation and their interaction is generic for solitons of saturable nonlinearity. In particular, they may fuse or "give birth" to new solitons upon collision.     

Conservative and dissipative fiber Bragg solitons (a review)

We present a comparative review of two classes of optical solitons—conservative and dissipative solitons—propagating in single-mode optical fibers in which refractive-index gratings are induced such that their period is comparable with the radiation wavelength. Fibers that have the Kerr nonlinearity and negligibly small losses and that do not gain radiation (conservative system) are described by traditional equations of the approximation of slowly varying amplitudes, and effects caused by the nonlinearity of the medium, such as nonlinear switching, optical bistability, and formation of conservative Bragg solitons are considered. It is shown that the passage beyond the scope of the approximation of slowly varying amplitudes makes it possible to describe new important effects, including localization of soliton centers near maxima of the refractive-index grating. Bright and dark conservative solitons are demonstrated, which are formed when the Kerr nonlinearity is replaced by the nonlinearity of two-level atomic systems. The properties of conservative solitons in resonance semiconductor Bragg structures with quantum wells are considered. Results of experimental studies of nonlinear effects in fibers with Bragg gratings are presented. For an active single-mode fiber with a Bragg refractive-index grating and nonlinear gain and absorption, dissipative solitons are described using the approximation of slowly varying amplitudes and inertialess nonlinearity. It is shown that the dissipative factors qualitatively change the properties of solitons compared to the conservative case. Using the Maxwell-Bloch equations, it is demonstrated that the ratio between the gain and absorption relaxation times significantly affects the stability of localized structures. The interaction of dissipative optical Bragg solitons is described. It is shown that, beyond the scope of the approximation of slowly varying amplitudes, the average velocity of propagating dissipative Bragg solitons acquires only discrete values, and formation of pairs of solitons with two values of the phase difference becomes possible. For a birefringent fiber, dissipative vector optical Bragg solitons are demonstrated.     

Circularly polarized optical spatial solitons

We present, experimentally and theoretically, a polymer material system that supports optical spatial solitons of circular polarization. We demonstrate one-dimensional circularly polarized dark solitons supported by photoisomerization nonlinearity in a bulk polymer.     

The evolution of two-frequency solitons in an optical fiber with a longitudinally nonuniform nonlinearity

The evolution of two-frequency solitons in an optical fiber, as well as the practically important special case of absence of the second-harmonic wave, in the presence of a longitudinal nonuniformity of the coefficients characterizing the propagation nonlinearity are considered. The solitons found for media with constant values of the nonlinearity coefficients are used as initial distributions for media with a periodic dependence of the nonlinearity coefficients on the longitudinal coordinate. Modulation of the coefficient of cubic or quadratic nonlinearity is shown to result in oscillations of the peak intensity of the solitons (in both their components if two-color solitons are considered). In the case of a weak modulation of the nonlinearity coefficients, oscillations of the peak intensity occur at the frequency coinciding with the frequency of modulation of the nonlinearity coefficients. Under the weak influence of a periodically modulated cubic nonlinearity, parameters of quadratic solitons also oscillate upon the propagation. Regions of stability of solitons in the space of the modulation parameters are established.     

Stable high-dimensional solitons in nonlocal competing cubic-quintic nonlinear media

We find and stabilize high-dimensional dipole and quadrupole solitons in nonlocal competing cubic-quintic nonlinear media. By adjusting the propagation constant, cubic, and quintic nonlinear coefficients, the stable intervals for dipole and quadrupole solitons that are parallel to the x-axis and those after rotating 45° counterclockwise around the origin of coordinate are found. For the dipole solitons and those after rotation, their stability is controlled by the propagation constant, the coefficients of cubic and quintic nonlinearity. The stability of quadrupole solitons is controlled by the propagation constant and the coefficient of cubic nonlinearity, rather than the coefficient of quintic nonlinearity, though there is a small effect of the quintic nonlinear coefficient on the stability. Our proposal may provide a way to generate and stabilize some novel high-dimensional nonlinear modes in a nonlocal system.     

Stability of Bragg grating solitons in a cubic-quintic nonlinear medium with dispersive reflectivity

We investigate the existence and stability of Bragg grating solitons in a cubic-quintic medium with dispersive reflectivity. It is found that the model supports two disjoint families of solitons. One family can be viewed as the generalization of the Bragg grating solitons in Kerr nonlinearity with dispersive reflectivity. On the other hand, the quintic nonlinearity is dominant in the other family. Stability regions are identified by means of systematic numerical stability analysis. In the case of the first family, the size of the stability region increases up to moderate values of dispersive reflectivity. However for the second family (i.e. region where quintic nonlinearity dominates), the size of the stability region increases even for strong dispersive reflectivity. For all values of m, there exists a subset of the unstable solitons belonging to the first family for which the instability development leads to deformation and subsequent splitting of the soliton into two moving solitons with different amplitudes and velocities.     

Localized Spatial Soliton Excitations in (2+1)-Dimensional NonlinearSchrödinger Equation with Variable Nonlinearity and an ExternalPotential

We report on the localized spatial soliton excitations in the multidimensional nonlinear Schrödinger equation with radially variable nonlinearity coefficient and an external potential. By using Hirota's binary differential operators, we determine a variety of external potentials and nonlinearity coefficients that can support nonlinear localized solutions of different but desired forms. For some specific external potentials and nonlinearity coefficients, we discuss features of the corresponding (2+1)-dimensional multisolitonic solutions, including ring solitons, lump solitons, and soliton clusters.     

Optical solitons supported by finite waveguide lattices with diffusive nonlocal nonlinearity

We investigate the properties of fundamental, multi-peak, and multi-peaked twisted solitons in three types of finite waveguide lattices imprinted in photorefractive media with asymmetrical diffusion nonlinearity. Two opposite soliton self-bending signals are considered for different families of solitons. Power thresholdless fundamental and multi-peaked solitons are stable in the low power region. The existence domain of two-peaked twisted solitons can be changed by the soliton self-bending signals. When solitons tend to self-bend toward the waveguide lattice, stable two-peaked twisted solitons can be found in a larger region in the middle of their existence region. Three-peaked twisted solitons are stable in the lower (upper) cutoff region for a shallow (deep) lattice depth. Our results provide an effective guidance for revealing the soliton characteristics supported by a finite waveguide lattice with diffusive nonlocal nonlinearity.     

Incoherent spatial optical solitons in photorefractive crystals

In this paper we have presented a theoretical investigation on the propagation properties of incoherent solitons in photorefractive media which is characterized by noninstantaneous saturating nonlinearity. Using mutual coherence function approach, we have obtained the equation of existence curve for such solitons. We have discussed the coherence characteristics of these solitons. We have found that solitons with a particular radius can possess two different critical powers. The dynamical evolution of these solitons has been discussed in detail by both analytical and numerical simulation.     

Dark Solitons for the Defocusing Cubic Nonlinear Schrödinger Equation with the Spatially Periodic Potential and Nonlinearity

We study the existence of dark solitons of the defocusing cubic nonlinear Schrödinger (NLS) eqaution with the spatially-periodic potential and nonlinearity. Firstly, we propose six families of upper and lower solutions of the dynamical systems arising from the stationary defocusing NLS equation. Secondly, by regarding a dark soliton as a heteroclinic orbit of the Poincaré map, we present some constraint conditions for the periodic potential and nonlinearity to show the existence of stationary dark solitons of the defocusing NLS equation for six different cases in terms of the theory of strict lower and upper solutions and the dynamics of planar homeomorphisms. Finally, we give the explicit dark solitons of the defocusing NLS equation with the chosen periodic potential and nonlinearity.