Bright solitons from defocusing nonlinearities

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.

Surface gap solitons in the Schrödinger equation with quintic nonlinearity and a lattice potential

We demonstrate the existence of surface gap solitons, a special type of asymmetric solitons, in the one-dimensional nonlinear Schrödinger equation with quintic nonlinearity and a periodic linear potential. The nonlinearity is suddenly switched in a step-like fashion in the middle of the transverse spatial region, while the periodic linear potential is chosen in the form of a simple sin 2 lattice. The asymmetric nonlinearities in this work can be realized by the Feshbach resonance in Bose-Einstein condensates or by the photorefractive effect in optics. The major peaks in the gap soliton families are asymmetric and they are located at the position of the jump in nonlinearity (at x = 0). In addition, the major peaks of the two-peak and multi-peak solitons at the position x = 0 are higher than those after that position, at x > 0. And such phenomena are more obvious when the value of chemical potential is large, or when the difference of nonlinearity values across the jump is big. Along the way, linear stability analysis of the surface gap solitons is performed and the stability domains are identified. It is found that in this model, the solitons in the first band gap are mostly stable (excepting narrow domains of instability at the edges of the gap), while those in the second band gap are mostly unstable (excepting extremely narrow domains of stability for fundamental solitons). These findings are also corroborated by direct numerical simulations.

Triangular bright solitons in nonlinear optics and Bose-Einstein condensates

We demonstrate what we believe to be novel triangular bright solitons that can be supported by the nonlinear Schrödinger equation with inhomogeneous Kerr-like nonlinearity and external harmonic potential, which can be realized in nonlinear optics and Bose-Einstein condensates. The profiles of these solitons are quite different from the common Gaussian or sech envelope beams, as their tops and bottoms are similar to the triangle and inverted triangle functions, respectively. The self-defocusing nonlinearity gives rise to the triangle-up solitons, while the self-focusing nonlinearity supports the triangle-down solitons. Here, we restrict our attention only to the lowest-order fundamental triangular solitons. All such solitons are stable, which is demonstrated by the linear stability analysis and also clarified by direct numerical simulations. In addition, the modulated propagation of both types of triangular solitons, with the modulated parameter being the strength of nonlinearity, is also presented. We find that such propagation is strongly affected by the form of the modulation of the nonlinearity. For example, the sudden change of the modulated parameter causes instabilities in the solitons, whereas the gradual variation generates stable solitons. Also, a periodic variation of the parameter causes the regular oscillation of solitons, with the same period. Interestingly, the triangle-up and triangle-down solitons can change into each other, when the parameter changes the sign.

Singular solitons

We demonstrate that the commonly known concept which treats solitons as nonsingular solutions produced by the interplay of nonlinear self-attraction and linear dispersion may be extended to include modes with a relatively weak singularity at the central point, which keeps their integral norm convergent. Such states are generated by self-repulsion, which should be strong enough, represented by septimal, quintic, and usual cubic terms in the framework of the one-, two-, and three-dimensional (1D, 2D, and 3D) nonlinear Schrödinger equations (NLSEs), respectively. Although such solutions seem counterintuitive, we demonstrate that they admit a straightforward interpretation as a result of screening of an additionally introduced attractive δ-functional potential by the defocusing nonlinearity. The strength ("bare charge") of the attractive potential is infinite in 1D, finite in 2D, and vanishingly small in 3D. Analytical asymptotics of the singular solitons at small and large distances are found, entire shapes of the solitons being produced in a numerical form. Complete stability of the singular modes is accurately predicted by the anti-Vakhitov-Kolokolov criterion (under the assumption that it applies to the model), as verified by means of numerical methods. In 2D, the NLSE with a quintic self-focusing term admits singular-soliton solutions with intrinsic vorticity too, but they are fully unstable. We also mention that dissipative singular solitons can be produced by the model with a complex coefficient in front of the nonlinear term.

Robust [Formula: see text] symmetry of two-dimensional fundamental and vortex solitons supported by spatially modulated nonlinearity

The real spectrum of bound states produced by [Formula: see text]-symmetric Hamiltonians usually suffers breakup at a critical value of the strength of gain-loss terms, i.e., imaginary part of the complex potential. The breakup essentially impedes the use of [Formula: see text]-symmetric systems for various applications. On the other hand, it is known that the [Formula: see text] symmetry can be made unbreakable in a one-dimensional (1D) model with self-defocusing nonlinearity whose strength grows fast enough from the center to periphery. The model is nonlinearizable, i.e., it does not have a linear spectrum, while the (unbreakable) [Formula: see text] symmetry in it is defined by spectra of continuous families of nonlinear self-trapped states (solitons). Here we report results for a 2D nonlinearizable model whose [Formula: see text] symmetry remains unbroken for arbitrarily large values of the gain-loss coefficient. Further, we introduce an extended 2D model with the imaginary part of potential ~xy in the Cartesian coordinates. The latter model is not a [Formula: see text]-symmetric one, but it also supports continuous families of self-trapped states, thus suggesting an extension of the concept of the [Formula: see text] symmetry. For both models, universal analytical forms are found for nonlinearizable tails of the 2D modes, and full exact solutions are produced for particular solitons, including ones with the unbreakable [Formula: see text] symmetry, while generic soliton families are found in a numerical form. The [Formula: see text]-symmetric system gives rise to generic families of stable single- and double-peak 2D solitons (including higher-order radial states of the single-peak solitons), as well as families of stable vortex solitons with m = 1, 2, and 3. In the model with imaginary potential ~xy, families of single- and multi-peak solitons and vortices are stable if the imaginary potential is subject to spatial confinement. In an elliptically deformed version of the latter model, an exact solution is found for vortex solitons with m = 1.

Purely Kerr nonlinear model admitting flat-top solitons

We elaborate one- and two-dimensional (1D and 2D) models of media with self-repulsive cubic nonlinearity, whose local strength is subject to spatial modulation that admits the existence of flat-top solitons of various types, including fundamental ones, 1D multipoles, and 2D vortices. Previously, solitons of this type were only produced by models with competing nonlinearities. The present setting may be implemented in optics and Bose-Einstein condensates. The 1D version gives rise to an exact analytical solution for stable flat-top solitons, and generic families may be predicted by means of the Thomas-Fermi approximation. Stability of the obtained flat-top solitons is analyzed by means of the linear-stability analysis and direct simulations. Fundamental solitons and 1D multipoles with k=1 and 2 nodes, as well as vortices with winding number m=1, are completely stable. For multipoles with k≥3 and vortices with m≥2, alternating stripes of stability and instability are identified in their parameter spaces.

Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity

Recent studies have demonstrated that defocusing cubic nonlinearity with local strength growing from the center to the periphery faster than r^{D}, in space of dimension D with radial coordinate r, supports a vast variety of robust bright solitons. In the framework of the same model, but with a weaker spatial-growth rate ∼r^{α} with α≤D, we test here the possibility to create stable localized continuous waves (LCWs) in one-dimensional (1D) and 2D geometries, localized dark solitons (LDSs) in one dimension, and localized dark vortices (LDVs) in two dimensions, which are all realized as loosely confined states with a divergent norm. Asymptotic tails of the solutions, which determine the divergence of the norm, are constructed in a universal analytical form by means of the Thomas-Fermi approximation (TFA). Global approximations for the LCWs, LDSs, and LDVs are constructed on the basis of interpolations between analytical approximations available far from (TFA) and close to the center. In particular, the interpolations for the 1D LDS, as well as for the 2D LDVs, are based on a deformed-tanh expression, which is suggested by the usual 1D dark-soliton solution. The analytical interpolations produce very accurate results, in comparison with numerical findings, for the 1D and 2D LCWs, 1D LDSs, and 2D LDVs with vorticity S=1. In addition to the 1D fundamental LDSs with the single notch and 2D vortices with S=1, higher-order LDSs with multiple notches are found too, as well as double LDVs, with S=2. Stability regions for the modes under consideration are identified by means of systematic simulations, the LCWs being completely stable in one and two dimensions, as they are ground states in the corresponding settings. Basic evolution scenarios are identified for those vortices that are unstable. The settings considered in this work may be implemented in nonlinear optics and in Bose-Einstein condensates.

Rotating vortex clusters in media with inhomogeneous defocusing nonlinearity

We show that media with inhomogeneous defocusing cubic nonlinearity growing toward the periphery can support a variety of stable vortex clusters nested in a common localized envelope. Nonrotating symmetric clusters are built from an even number of vortices with opposite topological charges, located at equal distances from the origin. Rotation makes the clusters strongly asymmetric, as the centrifugal force shifts some vortices to the periphery, while others approach the origin, depending on the topological charge. We obtain such asymmetric clusters as stationary states in the rotating coordinate frame, identify their existence domains, and show that the rotation may stabilize some of them.

Rotating and Precessing Dissipative-Optical-Topological-3D Solitons

We predict and study a new type of three-dimensional soliton: asymmetric rotating and precessing stable topological-dissipative-optical localized structures in homogeneous media with saturable amplification and absorption. The crucial factor determining their dynamics is the ratio of the diffusion coefficients characterizing the frequency dispersion and angular selectivity (dichroism) of the scheme. These vortex solitons exist and are stable for overcritical values of the selectivity coefficients and can be realized in lasers of large sizes with saturable absorption.

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.

Precession and nutation dynamics of nonlinearly coupled non-coaxial three-dimensional matter wave vortices

Nonlinearity is the driving force for numerous important effects in nature typically showing transitions between different regimes, regular, chaotic or catastrophic behavior. Localized nonlinear modes have been the focus of intense research in areas such as fluid and gas dynamics, photonics, atomic and solid state physics etc. Due to the richness of the behavior of nonlinear systems and due to the severe numerical demands of accurate three-dimensional (3D) numerical simulations presently only little knowledge is available on the dynamics of complex nonlinear modes in 3D. Here, we investigate the dynamics of 3D non-coaxial matter wave vortices that are trapped in a parabolic potential and interact via a repulsive nonlinearity. Our numerical simulations demonstrate the existence of an unexpected and fascinating nonlinear regime that starts immediately when the nonlinearity is switched-on and is characterized by a smooth dynamics representing torque-free precession with nutations. The reported motion is proven to be robust regarding various effects such as the number of particles, dissipation and trap deformations and thus should be observable in suitably designed experiments. Since our theoretical approach, i.e., coupled nonlinear Schrödinger equations, is quite generic, we expect that the obtained novel dynamical behavior should also exist in other nonlinear systems.

Stable Solitons in Three Dimensional Free Space without the Ground State: Self-Trapped Bose-Einstein Condensates with Spin-Orbit Coupling

By means of variational methods and systematic numerical analysis, we demonstrate the existence of metastable solitons in three dimensional (3D) free space, in the context of binary atomic condensates combining contact self-attraction and spin-orbit coupling, which can be engineered by available experimental techniques. Depending on the relative strength of the intra- and intercomponent attraction, the stable solitons feature a semivortex or mixed-mode structure. In spite of the fact that the local cubic self-attraction gives rise to the supercritical collapse in 3D, and hence the setting produces no true ground state, the solitons are stable against small perturbations, motion, and collisions.

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.

Solitons and vortices in two-dimensional discrete nonlinear Schrödinger systems with spatially modulated nonlinearity

We consider a two-dimensional (2D) generalization of a recently proposed model [Gligorić et al., Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anticontinuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual "extended" unstaggered bright solitons, in which all sites are excited in the AC limit, with the same sign across the lattice (they represent the most robust states supported by the lattice, their 1D counterparts being those considered as 1D bright solitons in the above-mentioned work), and the vortex cross, which is specific to the 2D setting. For all the existing states, we explore their stability (also analytically, when possible). Typical scenarios of instability development are exhibited through direct simulations.

Creation of vortices by torque in multidimensional media with inhomogeneous defocusing nonlinearity

Recently, a new class of nonlinear systems was introduced, in which the self-trapping of fundamental and vortical localized modes in space of dimension D is supported by cubic self-repulsion with a strength growing as a function of the distance from the center, r, at any rate faster that r^D. These systems support robust 2D and 3D modes which either do not exist or are unstable in other nonlinear systems. Here we demonstrate a possibility to create solitary vortices in this setting by applying a phase-imprinting torque to the ground state. Initially, a strong torque completely destroys the ground state. However, contrary to usual systems, where the destruction is irreversible, the present ones demonstrate a rapid restabilization and the creation of one or several shifted vortices orbiting the center. For the sake of comparison, we show analytically that, in the linear system with a 3D trapping potential, the action of a torque on the ground state is inefficient and creates only even-vorticity states with a small probability.

Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity

Toroidal modes in the form of so-called Hopfions, with two independent winding numbers, a hidden one (twist s), which characterizes a circular vortex thread embedded into a three-dimensional soliton, and the vorticity around the vertical axis (m), appear in many fields, including field theory, ferromagnetics, and semi- and superconductors. Such topological states are normally generated in multicomponent systems, or as trapped quasilinear modes in toroidal potentials. We uncover that stable solitons with this structure can be created, without any linear potential, in the single-component setting with the strength of repulsive nonlinearity growing fast enough from the center to the periphery, for both steep and smooth modulation profiles. Toroidal modes with s=1 and vorticity m=0, 1, 2 are produced. They are stable for m≤1, and do not exist for s>1. An approximate analytical solution is obtained for the twisted ring with s=1, m=0. Under the application of an external torque, it rotates like a solid ring. The setting can be implemented in a Bose-Einstein condensate (BEC) by means of the Feshbach resonance controlled by inhomogeneous magnetic fields.

Stabilization of solitons under competing nonlinearities by external potentials

We report results of the analysis for families of one-dimensional (1D) trapped solitons, created by competing self-focusing (SF) quintic and self-defocusing (SDF) cubic nonlinear terms. Two trapping potentials are considered, the harmonic-oscillator (HO) and delta-functional ones. The models apply to optical solitons in colloidal waveguides and other photonic media, and to matter-wave solitons in Bose-Einstein condensates loaded into a quasi-1D trap. For the HO potential, the results are obtained in an approximate form, using the variational and Thomas-Fermi approximations, and in a full numerical form, including the ground state and the first antisymmetric excited one. For the delta-functional attractive potential, the results are produced in a fully analytical form, and verified by means of numerical methods. Both exponentially localized solitons and weakly localized trapped modes are found for the delta-functional potential. The most essential conclusions concern the applicability of competing Vakhitov-Kolokolov (VK) and anti-VK criteria to the identification of the stability of solitons created under the action of the competing SF and SDF terms.

Bright solitons in nonlinear media with a self-defocusing double-well nonlinearity

We show that stable bright solitons can appear in a medium with spatially inhomogeneous self-defocusing (SDF) nonlinearity of a double-well structure. For a specific choice of the nonlinearity parameters, we obtain exact analytical solutions for the fundamental bright solitons. By making use of the linear stability analysis, the stability region in the parameter space for the exact fundamental bright soliton is obtained numerically. We also show the bifurcation from an antisymmetric to an asymmetric bright soliton for the SDF double-well nonlinearity.

Unbreakable PT symmetry of solitons supported by inhomogeneous defocusing nonlinearity

We consider bright solitons supported by a symmetric inhomogeneous defocusing nonlinearity growing rapidly enough toward the periphery of the medium, combined with an antisymmetric gain-loss profile. Despite the absence of any symmetric modulation of the linear refractive index, which is usually required to establish a parity-time (PT) symmetry in the form of a purely real spectrum of modes, we show that the PT symmetry is never broken in the present system, and that the system always supports stable bright solitons, i.e., fundamental and multi-pole ones. This fact is connected to the nonlinearizability of the underlying evolution equation. The increase of the gain-loss strength results, in lieu of the PT symmetry breaking, in merger of pairs of different soliton branches, such as fundamental and dipole, or tripole and quadrupole ones. The fundamental and dipole solitons remain stable at arbitrarily large values of the gain-loss coefficient.

Soliton gyroscopes in media with spatially growing repulsive nonlinearity

We find that the recently introduced model of self-trapping supported by a spatially growing strength of a repulsive nonlinearity gives rise to robust vortex-soliton tori, i.e., three-dimensional vortex solitons, with topological charges S≥1. The family with S=1 is completely stable, while the one with S=2 has alternating regions of stability and instability. The families are nearly exactly reproduced in an analytical form by the Thomas-Fermi approximation. Unstable states with S=2 and 3 split into persistently rotating pairs or triangles of unitary vortices. Application of a moderate torque to the vortex torus initiates a persistent precession mode, with the torus' axle moving along a conical surface. A strong torque heavily deforms the vortex solitons, but, nonetheless, they restore themselves with the axle oriented according to the vectorial addition of angular momenta.

Abundant soliton solutions of general nonlocal nonlinear Schrödinger system with external field

Periodic and quasi-periodic breather multi-solitons solutions, the dipole-type breather soliton solution, the rogue wave solution, and the fission soliton solution of the general nonlocal Schrödinger equation are derived by using the similarity transformation and manipulating the external potential function. The stability of the exact solitary wave solutions with the white noise perturbation also is investigated numerically.

Discrete localized modes supported by an inhomogeneous defocusing nonlinearity

We report that infinite and semi-infinite lattices with spatially inhomogeneous self-defocusing (SDF) onsite nonlinearity, whose strength increases rapidly enough toward the lattice periphery, support stable unstaggered (UnST) discrete bright solitons, which do not exist in lattices with the spatially uniform SDF nonlinearity. The UnST solitons coexist with stable staggered (ST) localized modes, which are always possible under the defocusing onsite nonlinearity. The results are obtained in a numerical form and also by means of variational approximation (VA). In the semi-infinite (truncated) system, some solutions for the UnST surface solitons are produced in an exact form. On the contrary to surface discrete solitons in uniform truncated lattices, the threshold value of the norm vanishes for the UnST solitons in the present system. Stability regions for the novel UnST solitons are identified. The same results imply the existence of ST discrete solitons in lattices with the spatially growing self-focusing nonlinearity, where such solitons cannot exist either if the nonlinearity is homogeneous. In addition, a lattice with the uniform onsite SDF nonlinearity and exponentially decaying intersite coupling is introduced and briefly considered. Via a similar mechanism, it may also support UnST discrete solitons. The results may be realized in arrayed optical waveguides and collisionally inhomogeneous Bose-Einstein condensates trapped in deep optical lattices. A generalization for a two-dimensional system is briefly considered.

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.

Solitary vortices supported by localized parametric gain

We demonstrate the existence and stability of bright vortex solitons sustained by a ring-shaped parametric gain, for both focusing and defocusing Kerr nonlinearities in lossy optical media. With the defocusing nonlinearity, the vortices are stable at all values of the detuning parameter, while under the focusing nonlinearity their stability region is limited to some positive values of the detuning. Unstable vortices in the focusing medium transform into stable rotating azimuthons.

Bright solitons in defocusing media with spatial modulation of the quintic nonlinearity

It has been recently demonstrated that self-defocusing (SDF) media with cubic nonlinearity, whose local coefficient grows from the center to the periphery fast enough, support stable bright solitons without the use of any linear potential. Our objective is to test the genericity of this mechanism for other nonlinearities, by applying it to one- and two-dimensional (1D and 2D) quintic SDF media. The models may be implemented in optics (in particular, in colloidal suspensions of nanoparticles), and the 1D model may be applied to the description of the Tonks-Girardeau gas of ultracold bosons. In 1D, the nonlinearity-modulation function is taken as g0+sinh2(βx). This model admits a subfamily of exact solutions for fundamental solitons. Generic soliton solutions are constructed in a numerical form and also by means of the Thomas-Fermi and variational approximations (TFA and VA). In particular, a new ansatz for the VA is proposed, in the form of "raised sech," which provides for an essentially better accuracy than the usual Gaussian ansatz. The stability of all the fundamental (nodeless) 1D solitons is established through the computation of the corresponding eigenvalues for small perturbations and also verified by direct simulations. Higher-order 1D solitons with two nodes have a limited stability region, all the modes with more than two nodes being unstable. It is concluded that the recently proposed inverted Vakhitov-Kolokolov stability criterion for fundamental bright solitons in systems with SDF nonlinearities holds here too. Particular exact solutions for 2D solitons are produced as well.

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.

Vortex solitons in defocusing media with spatially inhomogeneous nonlinearity

The analytical two- and three-dimensional vortex solitons with arbitrary values of vorticity are constructed in the cubic defocusing media with spatially inhomogeneous nonlinearity. The values of the nonlinearity coefficients are zero near the center and increase rapidly toward the periphery. In addition to the analytical ones, a number of vortex solitons are found numerically. It is shown that analytical vortex solitons are stable. Also, the stability region of the numerically constructed vortex solitons are given.

Spatial solitons under competing linear and nonlinear diffractions

We introduce a general model which augments the one-dimensional nonlinear Schrödinger (NLS) equation by nonlinear-diffraction terms competing with the linear diffraction. The new terms contain two irreducible parameters and admit a Hamiltonian representation in a form natural for optical media. The equation serves as a model for spatial solitons near the supercollimation point in nonlinear photonic crystals. In the framework of this model, a detailed analysis of the fundamental solitary waves is reported, including the variational approximation (VA), exact analytical results, and systematic numerical computations. The Vakhitov-Kolokolov (VK) criterion is used to precisely predict the stability border for the solitons, which is found in an exact analytical form, along with the largest total power (norm) that the waves may possess. Past a critical point, collapse effects are observed, caused by suitable perturbations. Interactions between two identical parallel solitary beams are explored by dint of direct numerical simulations. It is found that in-phase solitons merge into robust or collapsing pulsons, depending on the strength of the nonlinear diffraction.

Solitons supported by spatially inhomogeneous nonlinear losses

We uncover that, in contrast to the common belief, stable dissipative solitons exist in media with uniform gain in the presence of nonuniform cubic losses, whose local strength grows with coordinate η (in one dimension) faster than |η|. The spatially-inhomogeneous absorption also supports new types of solitons, that do not exist in uniform dissipative media. In particular, single-well absorption profiles give rise to spontaneous symmetry breaking of fundamental solitons in the presence of uniform focusing nonlinearity, while stable dipoles are supported by double-well absorption landscapes. Dipole solitons also feature symmetry breaking, but under defocusing nonlinearity.

Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities

We show that bimodal systems with a spatially nonuniform defocusing cubic nonlinearity, whose strength grows toward the periphery, can support stable two-component solitons. For a sufficiently strong cross-phase-modulation interaction, vector solitons with overlapping components become unstable, while stable families of solitons with spatially separated components emerge. Stable complexes with separated components may be built not only of fundamental solitons, but of multipoles, too.

Transfer and scattering of wave packets by a nonlinear trap

In the framework of a one-dimensional model with a tightly localized self-attractive nonlinearity, we study the formation and transfer (dragging) of a trapped mode by "nonlinear tweezers," as well as the scattering of coherent linear wave packets on the stationary localized nonlinearity. The use of a nonlinear trap for dragging allows one to pick up and transfer the relevant structures without grabbing surrounding "radiation." A stability border for the dragged modes is identified by means of analytical estimates and systematic simulations. In the framework of the scattering problem, the shares of trapped, reflected, and transmitted wave fields are found. Quasi-Airy stationary modes with a divergent norm, which may be dragged by a nonlinear trap moving at a constant acceleration, are briefly considered too.