We address the existence and properties of solitons in layered thermal media made of alternating focusing and defocusing layers. Such structures support robust bright solitons even if the averaged nonlinearity is defocusing. We show that non-oscillating solitons may form in any of the focusing domains, even in those located close to the sample edge, in contrast to uniform thermal media where light beams always oscillate when not launched exactly on the sample center. Stable multipole solitons may include more than four spots in layered media.
We study the dynamics of localized pulses in the complex cubic-quintic Ginzburg-Landau (GL) equation with strong nonlinearity management. The generalized complex GL equation, averaged over rapid modulations of the nonlinearity, is derived. We employ the variational approach to this averaged equation to derive a dynamical system for the dissipative soliton parameters. By using numerical simulations of this dynamical system and the full GP equation, we show that nonlinearity-managed dissipative solitons exist in the model. The parameter regions for stabilization of exploding soliton are obtained…
We predict that a photonic crystal fiber whose strands are filled with a defocusing nonlinear medium can support stable bright 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.
We have investigated mixed-gap vector solitons involving incoherently coupled fundamental and dipole components in a parity-time (PT) symmetric lattice with saturable nonlinearity. For the focusing case, vector solitons emerge from the semi-infinite and the first finite gaps, while for the defocusing case, vector solitons emerge from the first finite and the second finite gaps. For both cases, we find that stronger saturable nonlinearity is relative to sharper increase/decrease of soliton power with propagation constant and to narrower existence domain of vector solitons. This finding is helpf…
We address the interactions between optical solitons in the system with longitudinally varying nonlocality degree and nonlinearity strength. We consider a physical model describing light propagation in nematic liquid crystals featuring a strongly nonlocal nonlinear response. We reveal that the variation of the nonlocality and nonlinearity along the propagation direction can substantially enhance or weaken the interaction between out-of-phase solitons. This phenomenon manifests itself as a slowdown or acceleration of the soliton collision dynamics in one-dimensional geometries or of the soliton…
We show that an inhomogeneous defocusing nonlinearity that grows toward the periphery in the positive and negative transverse directions at different rates can support strongly asymmetric fundamental and multipole bright solitons, which are stable in wide parameter regions. In the limiting case when nonlinearity is uniform in one direction, solitons transform into stable domain walls (fronts), with constant or oscillating intensity in the homogeneous region, attached to a tail rapidly decaying in the direction of growing nonlinearity.
We investigate one-dimensional Helmholtz solitons with nonlocal nonlinearity. We show an exact analytical solution to the nonlocal nonlinear nonparaxial propagation equation in the cases of high and weak nonlocality. We also numerically find that the degree of nonlocality can affect the width of nonlocal Helmholtz soliton beams, but have no effect on their stability. Contrarily, nonparaxiality can affect their stability, but have no effect on their width.
We address necklace solitons supported by circular waveguide arrays with out-of-phase modulation of nonlinearity and linear refractive index. Such two-dimensional necklace solitons appear as rings of multiple out-of-phase bright spots. We show that necklace solitons are stable if their peak intensity is low or moderate and that the domain of stability shrinks with increasing nonlinearity modulation depth. However, we find that the stability domains expand with increasing number of necklace spots.
We put forward new properties of lattice solitons in materials and geometries where both, the linear refractive index and the nonlinearity are spatially modulated. We show that the interplay between linear and out-of-phase nonlinear refractive index modulations results in new soliton properties, including modifications of the soliton stability and transverse mobility, as well as shape transformations that may be controlled, e.g., by varying the light intensity.
We introduce multi-soliton sets in the two-dimensional medium with the second-harmonic-generating nonlinearity subject to spatial modulation in the form of a triangle of singular peaks. Various families of symmetric and asymmetric sets are constructed, and their stability is investigated. Stable symmetric patterns may be built of 1, 4, or 7 individual solitons, while stable asymmetric ones contain 1, 2, or 3 solitons. Symmetric and asymmetric patterns may demonstrate mutual bistability. The shift of the asymmetric single-soliton state from the central position is accurately predicted analytica…
We study the symmetric collisions of two mobile breathers/solitons in a model for coupled wave guides with a saturable nonlinearity. The saturability allows the existence of breathers with high power. Three main regimes are observed: breather fusion, breather reflection and breather creation. The last regime seems to be exclusive of systems with a saturable nonlinearity, and has been previously observed in continuous models. In some cases a ``symmetry breaking'' can be observed, which we show to be an numerical artifact.
The existence and stability of defect solitons in parity-time (PT) symmetric optical lattices with nonlocal nonlinearity are reported. It is found that nonlocality can expand the stability region of defect solitons. For positive or zero defects, fundamental and dipole solitons can exist stably in the semi-infinite gap and the first gap, respectively. For negative defects, fundamental solitons can be stable in both the semi-infinite gap and the first gap, whereas dipole solitons are unstable in the first gap. There exist a maximum degree of nonlocal nonlinearity, above which the fundamental sol…
We consider the impact of anisotropic nonlocality on the arrest of the collapse and stabilization of dipole-mode (DM) solitons in two-dimensional (2D) models of optical media with the diffusive nonlinearity. The nonlocal nonlinearity is made anisotropic through elliptic diffusivity. The medium becomes semi-local in the limit case of 1D diffusivity. Families of fundamental and DM solitons are found by means of the variational approximation (VA) and in a numerical form. We demonstrate that the collapse of 2D beams is arrested even in the semi-local system. The anisotropic nonlocality readily sta…
We address the properties of two-dimensional surface solitons supported by the interface of a waveguide array whose nonlinearity is periodically modulated. When the nonlinearity strength reaches its minima at the points where the linear refractive index attains its maxima, we found that nonlinear surface waves exist and can be made stable only within a limited band of input energy flows, and for lattice depths exceeding a lower threshold.
The existence and stability of stable bright solitons in one-dimensional (1D) media with a spatially periodical modulated Kerr nonlinearity are demonstrated by means of the linear-stability analysis and in direct numerical simulations. The nonlinear potential landscape can balance the fractional-order diffraction and thus stabilizes the solitons, making the model unique and governed by the recently introduced fractional Schrödinger equation with a self-focusing cubic nonlinear lattice. Both 1D fundamental and multihump solitons (in forms of dipole and tripole ones) are found, which occupy one …
We address the properties of optical solitons in thermal nonlinear media with a local refractive index defect that is capable to trap solitons launched even close to the sample boundary despite the boundary-mediated forces that tend to deflect all beams toward the center of the sample. We show that while such forces become more pronounced with increasing the input beam power the defect can trap only light below a critical power above which solitons are ejected. The dynamics of soliton ejection and the subsequent propagation may be controlled.
We report results for solitons in models of waveguides with focusing or defocusing saturable nonlinearity and a parity-time(PT )-symmetric complex-valued external potential of the Scarf-II type. The model applies to the nonlinear wave propagation in graded-index optical waveguides with balanced gain and loss. We find both fundamental and multipole solitons for both focusing and defocusing signs of the saturable nonlinearity in such PT -symmetric waveguides. The dependence of the propagation constant on the soliton's power is presented for different strengths of the nonlinearity saturation, S. …
We introduce a model of one- and two-dimensional (1D and 2D) optical media with the $χ^{(2)}$ nonlinearity whose local strength is subject to cusp-shaped spatial modulation, $χ^{(2)}\sim r^{-α}$, with $α>0$, which can be induced by spatially nonuniform poling. Using analytical and numerical methods, we demonstrate that this setting supports 1D and 2D fundamental solitons, at $α<1$ and $α<2$, respectively. The 1D solitons have a small instability region, while the 2D solitons have a stability region at $α<0.5$ and are unstable at $α>0.5$. 2D solitary vortices are found too. They are unstable, s…
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 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, fundamental and multi-pole ones. Such phenomenon is connected to non-linearizability of the…
We construct families of one-dimensional (1D) stable solitons in two-component $\mathcal{PT}$-symmetric systems with spin-orbit coupling (SOC) and quintic nonlinearity, which plays the critical role in 1D setups. The system models light propagation in a dual-core waveguide with skewed coupling between the cores. Stability regions for the solitons are identified in the system's parameter space. They include the main semi-infinite gap, and an additional finite $\textit{annex gap}$. Stability boundaries are identified by means of simulations of the perturbed evolution, which agree with results pr…
Dark solitons and localized defect modes against periodic backgrounds are considered in arrays of waveguides with defocusing Kerr nonlinearity constituting a nonlinear lattice. Bright defect modes are supported by local increase of the nonlinearity, while dark defect modes are supported by a local decrease of the nonlinearity. Dark solitons exist for both types of the defect, although in the case of weak nonlinearity they feature side bright humps making the total energy propagating through the system larger than the energy transferred by the constant background. All considered defect modes ar…
We address the properties of optical solitons that form in media with competing cubic-quintic nonlinearity and parity-time(PT)-symmetric complex-valued external potentials. The model describes the propagation of solitons in nonlinear optical waveguides with balanced gain and loss. We study the existence, stability, and robustness of fundamental, dipole, and multipole stationary solutions in this PT-symmetric system. The corresponding eigenvalue spectra diagrams for fundamental, dipole, tripole, and quadrupole solitons are presented. We show that the eigenvalue spectra diagrams for fundamental …
New two-component soliton solutions of the coupled high-frequency (HF) - low-frequency (LF) system, based on Schrödinger - Korteweg - de Vries (KdV) system with the Zakharov's coupling, are obtained for arbitrary relative strengths of the nonlinearity and dispersion in the LF component. The complex HF field is governed by the linear Schrödinger equation with a potential generated by the real LF component, which, in turn, is governed by the KdV equation including the ponderomotive coupling term, representing the feedback of the HF field onto the LF component. First, we study the evolution of pu…
We demonstrate existence of waves localized at the interface of two nonlinear periodic media with different coefficients of the cubic nonlinearity via the one-dimensional Gross--Pitaevsky equation. We call these waves the surface gap solitons (SGS). In the case of smooth symmetric periodic potentials, we study analytically bifurcations of SGS's from standard gap solitons and determine numerically the maximal jump of the nonlinearity coefficient allowing for the SGS existence. We show that the maximal jump vanishes near the thresholds of bifurcations of gap solitons. In the case of continuous p…
We present a numerical investigation of the propagation dynamics of a truncated Airy pulse in a highly dispersive and nonlinear medium by employing the split-step Fourier transform method and look, in particular, into the effects of fourth order dispersion (FOD) and cubic-quintic-septic nonlinearity on pulse evolution. Presence of FOD cancels the Airy pulse self-acceleration along with eclipsing the oscillatory tail during propagation in the linear regime. Further, we observe soliton shedding at low input pulse power in the presence of cubic and quintic nonlinearity and negative FOD. The emerg…
We investigate the switching dynamics in a $\mathcal{PT}$-symmetric fiber coupler composed of a saturable nonlinear material as the core. In such a saturable nonlinear medium, bistable solitons may evolve due to the balance between dispersion and saturable nonlinearity, which we extend in the context of the $\mathcal{PT}$-symmetric coupler. Our investigations of power-controlled and phase-sensitive switching show richer soliton switching dynamics than the currently existing conventional counterparts, which may lead to ultrafast and efficient all-optical switching dynamics at very low power owi…
We introduce one- and two-dimensional (1D and 2D) models of parity-time ($% \mathcal{PT}$) -symmetric couplers with the mutually balanced linear gain and loss applied to the two cores, and cubic-quintic (CQ) nonlinearity acting in each one. The 2D and 1D models may be realized in dual-core optical waveguides, in the spatiotemporal and spatial domains, respectively. Stationary solutions for $\mathcal{PT}$-symmetric solitons in these systems reduce to their counterparts in the usual coupler. The most essential problem is the stability of the solitons, which become unstable against symmetry break…
We observe experimentally higher-order solitons in waveguide arrays with defocusing saturable nonlinearity. Such solitons can comprise several in-phase bright spots and are stable above a critical power threshold. We elucidate the impact of the nonlinearity saturation on the domains of existence and stability of the observed complex soliton states.
We develop one numerical method to compute the optical vortex soliton with self-defocusing Kerr-type nonlocal nonlinearity. With the numerical simulation method, the propagation and interaction properties of such optical vortex solitons are investigated.
It is well known that the two-dimensional (2D) nonlinear Schrödinger equation (NLSE) with the cubic-quintic (CQ) nonlinearity supports a family of stable fundamental solitons, as well as solitary vortices (alias vortex rings), which are stable for sufficiently large values of the norm. We study stationary localized modes in a symmetric linearly coupled system of two such equations, focusing on asymmetric states. The model may describe "optical bullets" in dual-core nonlinear optical waveguides (including spatiotemporal vortices that were not discussed before), or a Bose-Einstein condensate (BE…
We address the properties of surface solitons supported by optical lattices imprinted in photorefractive media with asymmetric diffusion nonlinearity. Such solitons exist only in finite gaps of lattice spectrum. In contrast to latticeless geometries, where surface waves exist only when nonlinearity deflects light towards the material surface, the surface lattice solitons exist in settings where diffusion would cause beam bending against the surface.
It is known that optical-lattice (OL) potentials can stabilize solitons and solitary vortices against the critical collapse, generated by the cubic attractive nonlinearity in the 2D geometry. We demonstrate OLs can also stabilize various species of fundamental and vortical solitons against the super-critical collapse, driven by the double-attractive cubic-quintic nonlinearity (however, solitons remain unstable in the case of the pure quintic nonlinearity). Two types of OLs are considered, producing similar results: the 2D Kronig-Penney "checkerboard", and the sinusoidal potential. Soliton fami…
We study the generation of dissipative solitons (DSs) in the model of the fiber-laser cavities under the combined action of cubic-quintic nonlinearity, multiphoton absorption and/or multiphoton emission (nonlinear gain) and gain dispersion. A random component of the group-velocity dispersion (GVD) is included too. The DS creation and propagation is studied by means of a variational approximation and direct simulations, which are found to be in reasonable agreement. With a proper choice of the gain, robust DS operation regimes are predicted for different combinations of multiphoton absorption a…
We analyze a system of three two-dimensional nonlinear Schrödinger equations coupled by linear terms and with the cubic-quintic (focusing-defocusing) nonlinearity. We consider two versions of the model: conservative and parity-time ($\mathcal{PT}$) symmetric. These models describe triple-core nonlinear optical waveguides, with balanced gain and losses in the $\mathcal{PT}$-symmetric case. We obtain families of soliton solutions and discuss their stability. The latter study is performed using a linear stability analysis and checked with direct numerical simulations of the evolutional system of …
We study effects of nonlocality of the cubic self-focusing nonlinearity on the stability and symmetry-breaking bifurcation (SBB) of solitons in the model of a planar dual-core optical waveg- uide with nonlocal (thermal) nonlinearity. In comparison with the well-known coupled systems with the local nonlinearity, the present setting is affected by the competition of different spatial scales, viz., the coupling length and correlation radius of the nonlocality, $\sqrt{d}$. By means of numer- ical methods and variational approximation (VA, which is relevant for small d), we find that, with the incr…
We demonstrate the existence of two species of stable bright solitons, fundamental and dipole ones, in one-dimensional self-defocusing nonlocal media, with the local value of nonlinearity coefficient having one or several minima and growing at any rate faster than |x| at large values of coordinate x. The model can be derived for a slab optical waveguide with thermal nonlinearity. The most essential difference from the local counterpart of this system is the competition between two different spatial scales, the one determining the modulation pattern of the nonlinearity coefficient, and the corr…
Nonlinearities in lattices with topological band structures can induce topological interfaces in the bulk of structures and give rise to bulk solitons in the topological bandgaps. Here we study a photonic Chern insulator with saturable nonlinearity and show the existence of topological bulk solitons. The fundamental bulk solitons exhibit as semi-vortex solitons, where only one pseudospin component has a nonzero vorticity. The bulk solitons have equal angular momentum at different valleys. This phenomenon is a direct outcome of the topology of the linear host lattice and the angular momentum ca…
We construct families of optical semi-discrete composite solitons (SDCSs), with one or two independent propagation constants, supported by a planar slab waveguide, XPM-coupled to a periodic array of stripes. Both structures feature the cubic nonlinearity and support intrinsic modes with mutually orthogonal polarizations. We report three species of SDCSs, odd, even, and twisted ones, the first type being stable. Transverse motion of phase-tilted solitons, with potential applications to beam steering, is considered too.
An overview is given of basic models combining discreteness in their linear parts (i.e. the models are built as dynamical lattices) and nonlinearity acting at sites of the lattices or between the sites. The considered systems include the Toda and Frenkel-Kontorova lattices (including their dissipative versions), as well as equations of the discrete nonlinear Schroedinger (DNLS) and Ablowitz-Ladik (AL) types, and DNLS-AL combination in the form of the Salerno model. The interplay of discreteness and nonlinearity gives rise to a variety of states, most important ones being self-trapped discrete …
We present a series of chirp-free and chirped analytical nonautonomous soliton solutions to the generalized nonlinear Schrodinger equation (NLSE) with distributed coefficients by Darboux transformation from a trivial seed. For chirpfree nonautonomous soliton, the dispersion management term can change the motion of nonautonomous soliton and do not affect its shape at all. Especially,the classical optical soliton can be presented with variable dispersion term and nonlinearity when there is no gain. For chirped nonautonomous soliton, dispersion management can affect the shape and motion of nonaut…
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 address the existence and stability of two-dimensional solitons in optical or matter-wave media, which are supported by purely nonlinear lattices in the form of a periodic array of cylinders with self-focusing nonlinearity, embedded into a linear material. We show that such lattices can stabilize two-dimensional solitons against collapse. We also found that stable multipoles and vortex solitons are also supported by the nonlinear lattices, provided that the nonlinearity exhibits saturation.
We address the generation and interaction of dispersion-managed dissipative solitons (DMDS) in a model of fiber lasers with the cubic-quintic nonlinearity, multiphoton absorption and gain dispersion. Both anomalous and normal segments of the dispersion map include random dispersion fluctuations. Effects of the gain dispersion, higher-order nonlinearity and randomness on the generation of DMDS are demonstrated. The solitons exhibit breather-like evolution, and are found to be robust, up to a certain critical level of the random-dispersion component, which is sufficiently high. The roles of mult…
We demonstrate that spatially inhomogeneous defocusing nonlinear landscapes with the nonlinearity coefficient growing toward the periphery as [1+abs(r)]**a 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., a>D. Fundamental solitons are always stable, while multipoles and vortices are stable if the nonlinearity growth rate is large enough.
We present eight types of spatial optical solitons which are possible in a model of a planar waveguide that includes a dual-channel trapping structure and competing (cubic-quintic) nonlinearity. Among the families of trapped beams are symmetric and antisymmetric solitons of "broad" and "narrow" types, composite states, built as combinations of broad and narrow beams with identical or opposite signs ("unipolar" and "bipolar" states, respectively), and "single-sided" broad and narrow beams trapped, essentially, in a single channel. The stability of the families is investigated via eigenvalues of…
One-dimensional topological pumping of matter waves in two overlaid optical lattices moving with respect to each other is considered in the presence of attractive nonlinearity. It is shown that there exists a threshold nonlinearity level above which the matter transfer is completely arrested. Below this threshold the transfer of both dispersive wavepackets and solitons occurs in accordance with the predictions of the linear theory, i.e. it is quantized and determined by the dynamical Chern numbers of the lowest band. The breakdown of the transport is also explained by nontrivial topology of th…
We report a novel spontaneous symmetry breaking phenomenon and ghost states existed in the framework of the fractional nonlinear Schrödinger (FNLS) equation with focusing saturable nonlinearity and PT-symmetric potential. The continuous asymmetric soliton branch bifurcates from the fundamental symmetric one as the power exceeds some critical value. Intriguingly, the symmetry of fundamental solitons is broken into two branches of asymmetry solitons (alias ghost states) with complex conjugate propagation constants, which is solely in fractional media. Besides, the dipole (antisymmetry) and tripo…
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. 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 (TFA). Unstable states with S=2 and 3 split into persistently rotating pairs or triangles of unitary vortices. Application of a moderate torq…
A model of the optical media with a spatially structured Kerr nonlinearity is introduced. The nonlinearity strength is modulated by a set of singular peaks on top of a self-focusing or defocusing uniform background. The peaks may include a repulsive or attractive linear potential too. We find that a pair of mutually symmetric peaks readily gives rise to the spontaneous symmetry breaking (SSB) of modes pinned to individual peaks, while antisymmetric pinned modes are always unstable, transforming into robust spatially odd breathers. Three- and five-peak structures support symmetric modes, with i…
The real spectrum of bound states produced by PT-symmetric Hamiltonians usually suffers breakup at a critical value of the strength of gain-loss terms, i.e., imaginary part of the complex potential. On the other hand, it is known that the PT-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) PT symmetry in it is defined by spectra of continuous families of nonlinear self-trapped states (sol…
We demonstrate the generation of vortex solitons in a model of dissipative optical media with the singular anti-cubic (AC) nonlinearity, by launching a vorticity-carrying Gaussian input into the medium modeled by the cubic-quintic complex Ginzburg-Landau equation with the additional AC term. The effect of the latter term on the beam propagation is investigated in detail. An analytical result is produced for the asymptotic form of fundamental and vortical solitons at r --> 0, which is determined by the AC term. Numerical simulations identify parameter domains which maintain stable dissipative s…
We introduce a setting based on the one-dimensional (1D) nonlinear Schroedinger equation (NLSE) with the self-focusing (SF) cubic term modulated by a singular function of the coordinate, |x|^{-a}. It may be additionally combined with the uniform self-defocusing (SDF) nonlinear background, and with a similar singular repulsive linear potential. The setting, which can be implemented in optics and BEC, aims to extend the general analysis of the existence and stability of solitons in NLSEs. Results for fundamental solitons are obtained analytically and verified numerically. The solitons feature a …
It is commonly known that stable bright solitons in periodic potentials, which represent gratings in photonics/plasmonics, or optical lattices in quantum gases, exist either in the spectral semi-infinite gap (SIG) or in finite bandgaps. Using numerical methods, we demonstrate that, under the action of the cubic self-focusing nonlinearity, defects in the form of "holes" in two-dimensional (2D) lattices support continuous families of 2D solitons \textit{embedded} into the first two Bloch bands of the respective linear spectrum, where solitons normally do not exist. The two families of the \texti…
It has been recently demonstrated that self-defocusing (SDF) media with the cubic nonlinearity, whose local coefficient grows from the center to 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…
