Stability Of Bright Solitons Research Articles

Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarff-II potentials.

We present a unified theoretical study of the bright solitons governed by self-focusing and defocusing nonlinear Schrödinger (NLS) equations with generalized parity-time- (PT) symmetric Scarff-II potentials. Particularly, a PT-symmetric k-wave-number Scarff-II potential and a multiwell Scarff-II potential are considered, respectively. For the k-wave-number Scarff-II potential, the parameter space can be divided into different regions, corresponding to unbroken and broken PT symmetry and the bright solitons for self-focusing and defocusing Kerr nonlinearities. For the multiwell Scarff-II potential the bright solitons can be obtained by using a periodically space-modulated Kerr nonlinearity. The linear stability of bright solitons with PT-symmetric k-wave-number and multiwell Scarff-II potentials is analyzed in detail using numerical simulations. Stable and unstable bright solitons are found in both regions of unbroken and broken PT symmetry due to the existence of the nonlinearity. Furthermore, the bright solitons in three-dimensional self-focusing and defocusing NLS equations with a generalized PT-symmetric Scarff-II potential are explored. This may have potential applications in the field of optical information transmission and processing based on optical solitons in nonlinear dissipative but PT-symmetric systems.

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.

Exact solutions for bright and dark solitons in spatially inhomogeneous nonlinearity

Abstract 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.

Self-similar propagation and asymptotic optical waves in nonlinear waveguides.

The properties of self-similar optical waves propagating in a tapered cubic-quintic nonlinear waveguide are investigated. Using a lens-type transformation we obtain the exact analytical self-similar solutions which describe the propagation of bright-shaped solitons, dark-shaped solitons, kink-shaped solitons, and antikink-shaped solitons. The stability of the solutions is examined by numerical simulations such that stable bright solitons are found. Beyond the exact analytical solutions, asymptotic optical waves are also found by employing a direct ansatz. These waves possess linear chirps and can propagate self-similarly. The possibility of controlling the shape of output asymptotic optical waves is demonstrated. The analytical results are confirmed by numerical simulations. Finally, we investigate the generation and propagation properties of self-similar optical waves in a quintic nonlinear medium.

Bright solitons in Bose–Einstein condensates with field-induced dipole moments

We introduce an effectively one-dimensional model of a bosonic gas of particles carrying collinear dipole moments which are induced by an external polarizing field with the strength periodically modulated along the coordinate, which gives rise to an effective nonlocal nonlinear lattice in the condensate. The existence, shape and stability of bright solitons, appearing in this model, are investigated by means of the variational approximation and numerical methods. The mobility of solitons and interactions between them are studied too.

Dissipative ion-acoustic solitary and shock waves in a plasma with superthermal electrons

The excitation of a nonlinear ion-acoustic (IA) wave packet in a viscous plasma consisting of superthermal electrons and cold ions is investigated. It is assumed that the superthermal components obey the Kappa distribution. The standard reductive perturbation technique is employed to obtain a cubic complex Ginzburg–Landau type equation, which governs the dynamics of the amplitude of IA wave packets in a viscous plasma. In a weakly dissipative regime, by means of a perturbation method adopted from non-integrable dynamical systems theory, explicit soliton solutions are obtained of the dissipative dynamical system, determined as fixed points of a two-dimensional system of evolution equations describing the soliton amplitude and velocity. The stability of bright solitons is addressed using adiabatic perturbation theory in a weakly dissipative case. In a general dissipation regime, a novel type of travelling solitary (dissipative solitons) and shock-type solutions are derived for an envelope equation. Moreover, the dependence of the solitary and shock wave characteristics on relevant physical parameters of the problem is studied. It is shown that dissipative solitons do exist, sustained by a mutual balance between loss and gain terms, in addition to the interplay between dispersion and nonlinearity. The dependence of solitary and shock wave velocities on the parameters of the system are investigated. To feature distinctive characteristics of a dissipative soliton, it is compared with the bright soliton solution of the nonlinear Schrödinger equation.

Two-dimensional intraband solitons in lattice potentials with local defects and self-focusing nonlinearity

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 \textit{embedded defect solitons} (EDSs) are found to be continuously linked by the branch of \textit{gap defect solitons} (GDSs) populating the first finite bandgap. Further, the EDS branch traversing the first band links the GDS family with the branch of regular defect-supported solitons populating the SIG. Thus, we construct a continuous chain of regular, embedded, and gap-mode solitons ("superfamily") threading the entire bandgap structure considered here. The EDSs are stable in the first Bloch band, and partly stable in the second. They exist with the norm exceeding a minimum value, hence they do not originate from linear defect modes. Further, we demonstrate that double, triple and quadruple lattice defects support stable dipole-mode solitons and vortices.

Solitary modes in nonlocal media with inhomogeneous self-repulsive nonlinearity

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 correlation length of the nonlocality. The competition is explicitly exhibited by analytically obtained asymptotic form of generic solutions. Particular exact solutions are found analytically, and full soliton families are constructed in a numerical form. The multi-channel settings, with two or three local minima of the nonlinearity coefficient, are considered here for the first time, for both local and nonlocal models of the present type. States with multiple solitons launched into different channels are stable if the spacing between them exceeds a certain minimum value. A regime of stable Josephson oscillations of solitons between parallel channels is reported too.

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.

Dissipative solitons and vortices in polariton Bose-Einstein condensates

We examine spatial localisation and dynamical stability of Bose-Einstein condensates of exciton-polaritons in microcavities under the condition of off-resonant spatially inhomogeneous optical pumping both with and without a harmonic trapping potential. We employ the open-dissipative Gross-Pitaevskii model for describing an incoherently pumped polariton condensate coupled to an exciton reservoir, and reveal that spatial localisation of the steady-state condensate occurs due to the effective self-trapping created by the polariton flows supported by the spatially inhomogeneous pump, regardless of the presence of the external potential. A ground state of the polariton condensate with repulsive interactions between the quasiparticles represents a dynamically stable bright dissipative soliton. We also investigate the conditions for sustaining spatially localised structures with non-zero angular momentum in the form of single-charge vortices.

Stability of solitary waves in random nonlocal nonlinear media

We consider the interplay between nonlocal nonlinearity and randomness for two different nonlinear Schr\"odinger models. We show that stability of bright solitons in presence of random perturbations increases dramatically with the nonlocality-induced finite correlation length of the noise in the transverse plane, by means of both numerical simulations and analytical estimates. In fact, solitons are practically insensitive to noise when the correlation length of the noise becomes comparable to the extent of the wave packet. We characterize soliton stability using two different criteria based on the evolution of the Hamiltonian of the soliton and its power. The first criterion allows us to estimate a time (or distance) over which the soliton preserves its form. The second criterion gives the life-time of the solitary wave packet in terms of its radiative power losses. We derive a simplified mean field approach which allows us to calculate the power loss analytically in the physically relevant case of weakly correlated noise, which in turn serves as a lower estimate of the life-time for correlated noise in general case.

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.

Unstaggered-staggered solitons in two-component discrete nonlinear Schrödinger lattices

We present stable bright solitons built of coupled unstaggered and staggered components in a symmetric system of two discrete nonlinear Schrödinger equations with the attractive self-phase-modulation nonlinearity, coupled by the repulsive cross-phase-modulation interaction. These mixed modes are of a "symbiotic" type, as each component in isolation may only carry ordinary unstaggered solitons. The results are obtained in an analytical form, using the variational and Thomas-Fermi approximations (VA and TFA), and the generalized Vakhitov-Kolokolov (VK) criterion for the evaluation of the stability. The analytical predictions are verified against numerical results. Almost all the symbiotic solitons are predicted by the VA quite accurately and are stable. Close to a boundary of the existence region of the solitons (which may feature several connected branches), there are broad solitons which are not well approximated by the VA and are unstable.

Effects of temperature on the dynamic evolution of two-photon photorefractive screening spatial solitons

We investigate theoretically the temperature effects on the evolutions of both bright and dark screening spatial solitons in biased two-photon photorefractive crystals. For a stable bright or dark two-photon screening spatial soliton originally formed in a crystal at a given temperature, when the crystal temperature changes, it will evolve into another stable screening soliton if the temperature change is quite small, while it will become unstable or break down if the temperature change is large enough. The spatial shape of a stable two-photon screening spatial soliton can be changed by appropriately adjusting the crystal temperature.

Symbolic computation of solitons in the normal dispersion regime of inhomogeneous optical fibres

A nonlinear Schrödinger equation with varying dispersion, nonlinearity and gain (or absorption) is studied for ultrashort optical pulses propagating in inhomogeneous optical fibres in the case of normal dispersion. Using the modified Hirota method and symbolic computation, the bilinear form and analytic soliton solution are derived. Stable bright and dark solitons are observed in the normal dispersion regime. A periodically varying soliton and compressed soliton without any fluctuation are obtained. Combined and kink-shaped solitons are observed. Possibly applicable soliton control techniques, which are used to design dispersion-managed systems, are proposed. The proposed techniques may find applications in soliton management communication links, soliton compression and soliton control.

Frequency characteristics of the dark and bright surface solitons at a nonlinear metamaterial interface

We investigate the generation and properties of dark and bright surface waves at the interface of a conventional medium and a metamaterial. It is shown that the stable dark and bright surface solitons can coexist even when the nonlinear Kerr coefficients of the two media are negative, completely different from those at the interface between two conventional media. Most importantly, both the dark and bright surface solitons exist only within a certain range of the effective refractive index of the metamaterial, greatly depending on the frequency of the electromagnetic wave propagating in it. Therefore, the generation conditions for surface solitons, and the properties of the generated surface solitons such as the peak positions and the intensity of the dark and bright surface solitons, can be controlled by adjusting the structure parameters of the metamaterial and/or the frequency of the electromagnetic wave.

Controlling the amplitude of soliton in a growing Bose–Einstein condensate by means of Feshbach resonance

By using Darboux transformation, this paper studies analytically the nonlinear dynamics of a one-dimensional growing Bose–Einstein condensate (BEC). It is shown that the growing model has an important effect on the amplitude of the soliton in the condensates. In the absence of the growing model, there exhibits the stable alternate bright solitons in the condensates. In the presence of the growing model, the obtained results show that the amplitude of the bright soliton decreases (increases) for the BEC growing coefficient Ω < 0 (Ω > 0). Furthermore, we propose experimental protocols to manipulate the amplitude of the bright soliton by varying the scattering length via the Feshbach resonance in the future experiment.

Stability of bright solitons in some physical systems

Dynamical systems described by the modified quintic complex Ginzburg Landau equation and its derivative forms are considered and the stability of their bright soliton solution is investigated numerically by means of the split-step Fourier method. Some discussions related to the way of ensuring the stability of this solution are presented.

Bright solitons and soliton trains in a fermion-fermion mixture

We use a time-dependent dynamical mean-field-hydrodynamic model to predict and study bright solitons in a degenerate fermion-fermion mixture in a quasi-one-dimensional cigar-shaped geometry using variational and numerical methods. Due to a strong Pauli-blocking repulsion among identical spin-polarized fermions at short distances there cannot be bright solitons for repulsive interspecies fermion-fermion interactions. However, stable bright solitons can be formed for a sufficiently attractive interspecies interaction. We perform a numerical stability analysis of these solitons and also demonstrate the formation of soliton trains. These fermionic solitons can be formed and studied in laboratory with present technology.