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 1D, the nonlinearity-modulation function is taken as $% g_{0}+\sinh ^{2}(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 $\mathrm{sech}$", 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 "anti-VK" stability criterion for fundamental bright solitons in systems with SDF nonlinearities holds here too. Particular exact solutions for 2D solitons are produced as well.
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 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 of 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…
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 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 …
We systematically investigate two types of localized states - one is the optical analogue of the quantum relativistic Jackiw-Rebbi states, and the other is the trivial localized state - in interfaced binary waveguide arrays in the presence of cubic-quintic nonlinearity. By using the shooting method we can exactly calculate the profiles of these nonlinear localized states. Like in the case with Kerr nonlinearity, we demonstrate that these nonlinear localized states in the case with cubic-quintic nonlinearity also have a distinguishing feature which is completely different from all other well-kn…
We investigate the effects of spin-orbit coupling and Zeeman splitting on the coupled density-spin dynamics and collapse of the Bose-Einstein condensate driven by the quintic self-attraction in the same- and cross-spin channels. The characteristic feature of the collapse is the decrease in the width as given by the participation ratio of the density rather than by the expectation values of the coordinate. Qualitative arguments and numerical simulations reveal the existence of a critical spin-orbit coupling strength which either prohibits or leads to the collapse, and its dependence on other pa…
The statics and dynamics of a stable, mobile three-dimensional (3D) spatiotemporal vortex light bullet in a cubic-quintic nonlinear medium with a focusing cubic nonlinearity above a critical value and any defocusing quintic nonlinearity is considered. The present study is based on an analytic variational approximation and a full numerical solution of the 3D nonlinear Schrödinger equation. The 3D vortex bullet can propagate with a constant velocity. Stability of the vortex bullet is established numerically and variationally. The collision between two vortex bullets moving along the angular mome…
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, $\sim r^{α}$ with $α\leq D$, we here test the possibility to create stable\textit{\ localized continuous waves} (LCWs) in one- and two-dimensional (1D and 2D) geometries, \textit{% localized dark solitons} (LDSs) in 1D, and \textit{localized dark vortices} (LDVs) in 2D,…
We address the existence and stability of vortex-soliton (VS) solutions of the fractional nonlinear Schrödinger equation (NLSE) with competing cubic-quintic nonlinearities and the Lévy index (fractionality) taking values 1 \leqα\leq2. Families of ring-shaped VSs with vorticities s = 1,2, and 3 are constructed in a numerical form. Unlike the usual two-dimensional NLSE (which corresponds to α = 2), in the fractional model VSs exist above a finite threshold value of the total power,P. Stability of the VS solutions is investigated for small perturbations governed by the linearized equation, and co…
In this paper, we propose a liquid core suspended photonic crystal fiber (LCSPCF) as a potential waveguide structure for nonlinear applications. We emphasize the dramatic improvement of the nonlinear properties of the PCF through the infiltration of liquid and highlight the effect of suspension in the further enhancement of nonlinearity. To demonstrate the capability of LCSPCF, we perform a modulation instability (MI) study and illustrate the MI enhancement in the LCSPCF against conventional PCF. As the LCSPCF possesses high nonlinearity, it is important to consider the effects of higher-order…
We construct families of fundamental, dipole, and tripole solitons in the fractional Schrödinger equation (FSE)\ incorporating self-focusing cubic and defocusing quintic terms modulated by factors $\cos ^{2}x$ and $\sin^{2}x$, respectively. While the fundamental solitons are similar to those in the model with the uniform nonlinearity, the multipole complexes exist only in the presence of the nonlinear lattice. The shapes and stability of all the solitons strongly depend on the Lévy index (LI)\ that determines the FSE fractionality. Stability areas are identified in the plane of LI and propagat…
We address the existence and stability of localized modes in the framework of the fractional nonlinear Schroedinger equation (FNSE) with the focusing cubic or focusing-defocusing cubic-quintic nonlinearity and a confining harmonic-oscillator (HO) potential. Approximate analytical solutions are obtained in the form of Hermite-Gauss modes. The linear stability analysis and direct simulations reveal that, under the action of the cubic self-focusing, the single-peak ground state and dipole mode are stabilized by the HO potential at values of the Levy index (the fractionality degree) alpha = 1 and …
Competing nonlinearities, such as the cubic (Kerr) and quintic nonlinear terms whose strengths are of opposite signs (the coefficients in front of the nonlinearities), exist in various physical media (in particular, in optical and matter-wave media). A benign competition between self-focusing cubic and self-defocusing quintic nonlinear nonlinearities (known as cubic-quintic model) plays an important role in creating and stabilizing the self-trapping of D-dimensional localized structures, in the contexts of standard nonlinear Schrödinger equation. We incorporate an external periodic potential (…
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 reveal the existence of continuous families of guided single-mode solitons in planar waveguides with weakly nonlinear active core and absorbing boundaries. Stable propagation of TE and TM-polarized solitons is accompanied by attenuation of all other modes, i.e. the waveguide features properties of conservative and dissipative systems. If the linear spectrum of the waveguide possesses exceptional points, which occurs in the case of TM-polarization, an originally focusing (defocusing) material nonlinearity may become effectively defocusing (focusing). This occurs due to the geometric phase of…
The nonlinear lattice---a new and nonlinear class of periodic potentials---was recently introduced to generate various nonlinear localized modes. Several attempts failed to stabilize two-dimensional (2D) solitons against their intrinsic critical collapse in Kerr media. Here, we provide a possibility for supporting 2D matter-wave solitons and vortices in an extended setting---the cubic and quintic model---by introducing another nonlinear lattice whose period is controllable and can be different from its cubic counterpart, to its quintic nonlinearity, therefore making a fully `nonlinear quasi-cr…
We observe and analyze formation, decay, and subsequent regeneration of ring-shaped clusters of (2+1)-dimensional spatial solitons (filaments) in a medium with the cubic-quintic (focusing-defocusing) self-interaction and strong dissipative nonlinearity. The cluster of filaments, that remains stable over ~17.5 Rayleigh lengths, is produced by the azimuthal modulational instability from a parent ring-shaped beam with embedded vorticity l = 1. In the course of still longer propagation, the stability of the soliton cluster is lost under the action of nonlinear losses. The annular cluster is then s…
We investigate numerically soliton-plasma interaction in a noble-gas-filled silica hollow-core anti-resonant fiber pumped in the mid-IR at 3.0 μm. We observe multiple soliton self-compression stages due to distinct stages where either the self-focusing or the self-defocusing nonlinearity dominates. Specifically, the parameters may be tuned so the competing plasma self-defocusing nonlinearity only dominates over the Kerr self-focusing nonlinearity around the soliton self-compression stage, where the increasing peak intensity on the leading pulse edge initiates a competing self-defocusing plasma…
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 demonstrate a possibility of the creation of stable optical solitons combining one continuous and one discrete coordinate, with embedded vorticity, in an array of planar waveguides with intrinsic cubic-quintic nonlinearity. The same system may be realized in terms of the spatiotemporal light propagation in an array of tunnel-coupled optical fibers with the cubic-quintic nonlinearity. In contrast with zero-vorticity states, semidiscrete vortex solitons do not exist without the quintic term in the nonlinearity. Two types of the solitons, \emph{viz.}, intersite- and onsite-centered ones (IC an…
We report results of a systematic numerical analysis of interactions between three-dimensional (3D) fundamental solitons, performed in the framework of the nonlinear Schrödinger equation (NLSE) with the cubic-quintic (CQ) nonlinearity, combining the self-focusing and defocusing terms. The 3D NLSE with the CQ terms may be realized in terms of spatiotemporal propagation of light in nonlinear optical media, and in Bose-Einstein condensates, provided that losses may be neglected. The first part of the work addresses interactions between identical fundamental solitons, with phase shift $% φ$ betwee…
