Cubic-quintic Research Articles

AN EFFICIENT APPROACH FOR SOLVING THE FRACTAL, DAMPED CUBIC–QUINTIC DUFFING’S EQUATION

The main goal of this work is to focus on using He’s two-scale fractal dimension transform, the Caputo–Fabrizio fractional-order derivative, and the harmonic balance and the homotopy methods are applied for deriving the approximate solution of the fractal, damped cubic–quintic Duffing’s equation when the fractional derivative order of the inertia term is not twice of that of the damping term. Numerical results obtained from the derived expressions and the numerical integration solution show good agreement, especially at small values of the nonlinear parameters. Furthermore, when the fractal order of the damping term decreases, the damping oscillation frequency values increase with a decrease in the system wavelength values, which indicates a slower decay in the system oscillation amplitudes. Our solution procedures elucidate the applicability of He’s two-scale fractal dimension transform for solving nonlinear dynamic systems with inertia and damping fractal terms.

Spatiotemporal instabilities in [formula omitted]-symmetric periodic structures with non-Kerr fiber Bragg gratings

The influence of cubic–quintic (CQ) nonlinearity on the modulation instability (MI) in fiber Bragg grating structures in the conventional and three PT-symmetry types such as unbroken, singularity and broken are investigated. The MI gain was obtained through standard linear stability analysis. The effect of cubic nonlinearity combined with focusing and defocusing quintic nonlinearity on MI for both anomalous and normal GVD regimes, as well as the top and bottom edges of the photonic bandgap. We find that the MI gain varies dramatically because of the consequences of cubic–quintic nonlinearity. Finally, we show that even in the presence of PT-symmetry, effective control of the MI can be brought to light by adjusting the spatial instability parameter, cubic nonlinearity, focusing and defocusing quintic nonlinearity. The investigation yields a variety of spectrum types, including symmetry, asymmetry, monotonically increasing gain, sideband spectrum, broad spectrum, and so on. For the few physical variables, we observe an unusual spectrum that is more discrete in nature rather than continuous. After introducing the PT-symmetry, the emergence of solitons as in nonlinear periodic structure is also explored.

Symmetry-breaking bifurcations and ghost states in the fractional nonlinear Schrödinger equation with a PT-symmetric potential

We report symmetry-breaking and restoring bifurcations of solitons in a fractional Schrödinger equation with cubic or cubic-quintic (CQ) nonlinearity and a parity-time-symmetric potential, which may be realized in optical cavities. Solitons are destabilized at the bifurcation point, and, in the case of CQ nonlinearity, the stability is restored by an inverse bifurcation. Two mutually conjugate branches of ghost states (GSs), with complex propagation constants, are created by the bifurcation, solely in the case of fractional diffraction. While GSs are not true solutions, direct simulations confirm that their shapes and results of their stability analysis provide a "blueprint" for the evolution of genuine localized modes in the system.

Semidiscrete Vortex Solitons

A possibility of creation of stable optical solitons combining one continuous and one discrete coordinates, with embedded vorticity, in an array of planar waveguides with intrinsic cubic–quintic (CQ) nonlinearity is demonstrated. The same system may be realized in terms of the spatiotemporal light propagation in an array of tunnel‐coupled optical fibers with the CQ 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, viz., intersite‐centered (IC) and onsite‐centered (OC) ones, with even and odd numbers N of actually excited sites in the discrete direction, are identified. The modes carrying the embedded vorticity and 2 are considered. In accordance with their symmetry, the vortex solitons of the OC type exhibit an intrinsic core, whereas the IC solitons with small N may have a coreless structure. Facilitating their creation in the experiment, the modes reported in the present work may be much more compact states than their counterparts considered in other systems, and they feature strong anisotropy. They can be set in motion in the discrete direction, provided that the coupling constant exceeds a certain minimum value. Collisions between moving vortex solitons are also considered.

Cubic quintic Ginzburg Landau equation as a model for resonant interaction of EM field with nonlinear media

In this work, we investigate the possibility of approximating saturable nonlinearity, which is commonly used in complex Ginzburg–Landau equation (CGLE) for modelling resonant interaction of an electromagnetic field with nonlinear media, with cubic-quintic (CQ) nonlinearity. To validate the suggested approximations, we use variational method to estimate 2D analytical solutions of the CGLE with both saturable and CQ nonlinearity. The paper compares three ways to determine parameters of the CQ approximation and discusses the obtained results in terms of accuracy.

Interactions of three-dimensional solitons in the cubic-quintic model.

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 between them, separated by a finite distance in the free space. The outcome strongly changes with the variation of φ: in-phase solitons with φ = 0, or with sufficiently small φ, merge into a single fundamental soliton, with weak residual oscillations in it (in contrast to the merger into a strongly oscillating breather, which is exhibited by the 1D version of the same setting), while the choice of φ = π leads to fast separation between mutually repelling solitons. At intermediate values of φ, such as φ = π/2, the interaction is repulsive too, breaking the symmetry between the initially identical fundamental solitons, there appearing two solitons with different total energies (norms). The symmetry-breaking effect is qualitatively explained, similar to how it was done previously for 1D solitons. In the second part of the work, a pair of fundamental solitons trapped in a 2D potential is considered. It is demonstrated that they may form a slowly rotating robust "molecule," if initial kicks are applied to them in opposite directions, perpendicular to the line connecting their centers.

Analytic methods to find beating transitions of asymmetric Gaussian beams in GNLS equations.

In a simple model of propagation of asymmetric Gaussian beams in nonlinear waveguides, described by a reduction to ordinary differential equations of generalized nonlinear Schrödinger equations with cubic-quintic (CQ) and saturable (SAT) nonlinearities and a graded-index profile, the beam widths exhibit two different types of beating behavior, with transitions between them. We present an analytic model to explain these phenomena, which originate in a 1:1 resonance in a 2 degree-of-freedom Hamiltonian system. We show how small oscillations near a fixed point close to 1:1 resonance in such a system can be approximated using an integrable Hamiltonian and, ultimately, a single first order differential equation. In particular, the beating transitions can be located from coincidences of roots of a pair of quadratic equations, with coefficients determined (in a highly complex manner) by the internal parameters and initial conditions of the original system. The results of the analytic model agree with the numerics of the original system over large parameter ranges, and allow new predictions that can be verified directly. In the CQ case, we identify a band of beam energies for which there is only a single beating transition (as opposed to 0 or 2) as the eccentricity is increased. In the SAT case, we explain the sudden (dis)appearance of beating transitions for certain values of the other parameters as the grade-index is changed.

Propagation Dynamics of Finite-Energy Airy Pulses in a Cubic–Quintic Competing Nonlinear Fiber

We numerically simulated dynamic propagation of in-phase and out-of-phase finite-energy Airy pulses in anomalous regions of optical fibers and analyzed the effects of cubic–quintic (CQ) competing nonlinearity and separation parameter on evolution properties. Numerical results showed entirely different effects of quintic nonlinearity on in-phase and out-of-phase Airy pulses that vary with different separation parameters. For in-phase Airy pulses, a central quasi-soliton pulse is shed from all finite-energy Airy pulses when separation parameter is certain. CQ competing nonlinearity exhibits profound effects on propagation dynamics of soliton pulses. However, for out-of-phase Airy pulses, two quasi-soliton pulses and more dispersion wave pulses are symmetrically shed from the main lobe of out-of-phase Airy pulses. CQ competing nonlinearity slightly impacts evolution of out-of-phase Airy pulses depending on peak intensity, which varies with propagation distance and unlike in cases of in-phase Airy pulses.

Propagating properties of spatial solitons in the competing nonlocal cubic-quintic nonlinear media

We study the new spatial optical solitons and their propagating properties in the one-dimensional nonlocal cubic-quintic (C-Q) nonlinear model by the numerical method. We obtain multi-bright solitons and multipole soliton solutions in the one-dimensional nonlocal C-Q nonlinear model. The propagation of bright solitons is stable in the competing nonlocal cubic self-defocusing and quintic self-focusing nonlinear media when these nonlocal and nonlinear parameters are in the appropriate value domain. Considering the different nonlinear cubic effects, the interaction between two optical solitons with the same phase in the general nonlocal media displays the attraction or the repulsion for different nonlocal and nonlinear parameters. We find that the interval of two solitons affects the interaction between them. The refractive index is changed with the propagating constant when the nonlocal constant d3 is 10. Moreover, the triplepole, quadrupole and pentapole solitons can propagate steadily when the nonlocal parameters are appropriate, but hexa-pole (or above) solitons propagate unsteadily for any nonlocal parameter. Furthermore, we investigate the multi-pole solitons and their propagation stabilities by the Newton difference method and the Fourier split step method, obtain the stable propagation conditions for dipole, triplepole and quadrupole solitons, and find that the propagation of the pentapole and higher-order pole solitons is unstable. We also discuss the interactions of multi-pole solitons when they propagate along the axis z. The interactions are attraction or repulsion when the nonlocal and the nonlinear parameters are different. Meanwhile, we simulate the evolution of the refractive index along the axis z when the spatial optical solitons are multi-pole solitons. Finally, we study the relation between the power of soliton and the propagation constant under different degree of nonlocality. The power of the single bright soliton does not monotonically increase with the increasing propagation constant when the degree of nonlocality d3 is 10. We also derive the relation between the power of dipole bright solitons with the cubic nonlinearity parameter and the propagation constant under different degree of nonlocality. The power decreases monotonically with the increasing propagation constant when the cubic nonlinearity is a certain value or with the increasing cubic nonlinearity when the propagation constant is a certain value.

Building patterns by traveling dipoles and vortices in two-dimensional periodic dissipative media

We analyze pattern-formation scenarios in the two-dimensional (2D) complex Ginzburg–Landau (CGL) equation with the cubic–quintic (CQ) nonlinearity and a cellular potential. The equation models laser cavities with built-in gratings, which stabilize 2D patterns. The pattern-building process is initiated by kicking a compound mode, in the form of a dipole, quadrupole, or vortex which is composed of four local peaks. The hopping motion of the kicked mode through the cellular structure leads to the generation of various extended patterns pinned by the structure. In the ring-shaped system, the persisting freely moving dipole hits the stationary pattern from the opposite side, giving rise to several dynamical regimes, including periodic elastic collisions, i.e., persistent cycles of elastic collisions between the moving and quiescent dissipative solitons, and transient regimes featuring several collisions which end up by absorption of one soliton by the other. Still another noteworthy result is the transformation of a strongly kicked unstable vortex into a stably moving four-peaked cluster.

Stability boundary and collisions of two-dimensional solitons inPT-symmetric couplers with the cubic-quintic nonlinearity

We introduce one- and two-dimensional (1D and 2D) models of parity-time (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 wave guides in the spatiotemporal and spatial domains, respectively. Stationary solutions for 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 breaking with the increase of the energy (norm) and retrieve the stability at still larger energies. The boundary value of the intercore-coupling constant, above which the solitons are completely stable, is found by means of an analytical approximation, based on the cw (zero-dimensional) counterpart of the system. The approximation demonstrates good agreement with numerical findings for the 1D and 2D solitons. Numerical results for the stability limits of the 2D solitons are obtained by means of the computation of eigenvalues for small perturbations, and verified in direct simulations. Although large parts of the soliton families are unstable, the instability is quite weak. Collisions between 2D solitons in the PT-symmetric coupler are studied by means of simulations. Outcomes of the collisions are inelastic but not destructive, as they do not break the PT symmetry.

Suppressing the critical collapse of solitons by one-dimensional quintic nonlinear lattices

The stabilization of two-dimensional (2D) solitons in self-focusing Kerr media against the critical collapse with the help of nonlinear lattices (NLs), i.e., spatially periodic modulations of the local strength of the cubic nonlinearity, was recently investigated in detail. In one dimension (1D), the critical collapse is induced by the quintic self-focusing. Degenerate families of unstable localized modes which exist in these situations are usually called Townes solitons. We aim to explore a possibility of the stabilization of the 1D Townes solitons by means of NLs acting on the quintic or cubic–quintic (CQ) terms in the corresponding nonlinear-Schrödinger/Gross–Pitaevskii equation (NLSE/GPE). These settings may be realized in nonlinear optics and Bose–Einstein condensates (BECs). We develop the variational approximation (VA) for the CQ model, and use numerical methods to study the stability and mobility of solitons in both models, quintic and CQ. “Two-tier” and higher-order solitons, including dipole modes, are also found in the CQ medium (chiefly, in the case when the quintic nonlinearity tends to be self-defocusing). The stability region for fundamental solitons amounts to a narrow stripe in a parameter plane of the model with the quintic-only nonlinearity, being much broader in the case of the CQ nonlinearity.

Generation of polygonal soliton clusters and fundamental solitons in dissipative systems by necklace-ring beams with radial-azimuthal phase modulation

We demonstrate that, in a two-dimensional dissipative medium described by the cubic-quintic (CQ) complex Ginzburg-Landau (CGL) equation with the viscous (spectral-filtering) term, necklace rings carrying a mixed radial-azimuthal phase modulation can evolve into polygonal or quasipolygonal stable soliton clusters, and into stable fundamental solitons. The outcome of the evolution is controlled by the depth and azimuthal anharmonicity of the phase-modulation profile, or by the radius and number of "beads" in the initial necklace ring. Threshold characteristics of the evolution of the patterns are identified and explained. Parameter regions for the formation of the stable polygonal and quasipolygonal soliton clusters, and of stable fundamental solitons, are identified. The model with the CQ terms replaced by the full saturable nonlinearity produces essentially the same set of basic dynamical scenarios; hence this set is a universal one for the CGL models.

Stabilization of one-dimensional solitons against the critical collapse by quintic nonlinear lattices

It has been recently discovered that stabilization of two-dimensional (2D) solitons against the critical collapse in media with the cubic nonlinearity by means of nonlinear lattices (NLs) is a challenging problem. We address the 1D version of the problem, i.e., the nonlinear-Schr\"odinger equation (NLSE) with the quintic or cubic-quintic (CQ) terms, the coefficient in front of which is periodically modulated in space. The models may be realized in optics and Bose-Einstein condensates (BECs). Stability diagrams for the solitons are produced by means of numerical methods and analytical approximations. It is found that the sinusoidal NL stabilzes solitons supported by the quintic-only nonlinearity in a narrow stripe in the respective parameter plane, on the contrary to the case of the cubic nonlinearity in 2D, where the stabilization of solitons by smooth spatial modulations is not possible at all. The stability region is much broader in the 1D CQ model, where higher-order solitons may be stable too.

Ultrashort self-similar solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients in the inhomogeneous fiber

By means of the similarity transformation, we obtain exact self-similar solutions (similaritons) of the generalized cubic-quintic (CQ) nonlinear Schrödinger equation with spatially inhomogeneous group velocity dispersion, CQ nonlinearity and amplification or attenuation. Exact balance conditions between the dispersion, nonlinearity and the gain/loss have been obtained. As an example, we investigate their propagation dynamics in the dispersion decreasing fiber (DDF). Considering the fluctuation of the fiber parameter in real application, the exact balance conditions do not satisfy, and so we perform direct numerical analysis with initial 10% white noise for the bright similariton in both the DDF and the periodic distributed amplification system. Numerical calculations indicate stable propagation of the bright similariton over tens of dispersion lengths. These ultrashort self-similar optical waves are potentially useful for all-optical data-processing schemes and the design of beam compressors and amplifiers.

Splitting broad beams into arrays of dissipative spatial solitons by material and virtual gratings

We elaborate two generic methods for producing two-dimensional (2D) spatial soliton arrays (SSAs) in the framework of the cubic–quintic (CQ) complex Ginzburg–Landau (CGL) model. The first approach deals with a broad beam launched into the dissipative nonlinear medium which is equipped with an imprinted grating of a sufficiently sharp form. The beam splits into a cluster of jets, each subsequently self-trapping into a stable soliton, if the power is sufficient. We consider two kinds of sharp gratings—‘raised-cosine’ (RC) and Kronig–Penney (KP) lattices—and two types of input beams: fundamental and vortical. By selecting appropriate parameters, this method makes it possible to create various types of soliton arrays, such as solid, annular (with single and double rings) and cross-shaped ones. The second method uses a ‘virtual lattice’, in the form of a periodic transverse phase modulation imprinted into the broad beam which is passed through an appropriate phase mask and then shone into a uniform nonlinear medium. Two different types of masks are considered; in the form of a ‘checkerboard’ or ‘tilings’. In these cases, broad fundamental and vortical beams may also evolve into stable SSAs if the beam power and spacing of the virtual phase lattice are large enough. By means of the latter technique, square-shaped, hexagonal and quasi-crystalline SSAs can be created.

Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity

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 have not been discussed before), or a Bose–Einstein condensate (BEC) loaded into a “dual-pancake” trap. Each family of solutions in the single-component model has two different counterparts in the coupled system, one symmetric and one asymmetric. Similarly to the earlier studied coupled 1D system with the CQ nonlinearity, the present model features bifurcation loops, for fundamental and vortex solitons alike: with the increase of the total energy (norm), the symmetric solitons become unstable at a point of the direct bifurcation, which is followed, at larger values of the energy, by the reverse bifurcation restabilizing the symmetric solitons. However, on the contrary to the 1D system, both the direct and reverse bifurcation may be of the subcritical type, at sufficiently small values of the coupling constant, λ . Thus, the system demonstrates a double bistabilityfor the fundamental solitons. The stability of the solitons is investigated via the computation of instability growth rates for small perturbations. Vortex rings, which we study for two values of the “spin”, s = 1 and 2 , may be subject to the azimuthal instability, like in the single-component model. In particular, complete destabilization of asymmetric vortices is demonstrated for a sufficiently strong linear coupling. With the decrease of λ , a region of stable asymmetric vortices appears, and a single region of bistability for the vortices is found. We also develop a quasi-analytical approach to the description of the bifurcations diagrams, based on the variational approximation. Splitting of asymmetric vortices, induced by the azimuthal instability, is studied by means of direct simulations. Interactions between initially quiescent solitons of different types are studied too. In particular, we confirm the prediction of the reversal of the sign of the interaction (attractive/repulsive for in-phase/out-of-phase pairs) for the solitons with the odd spin, s = 1 , in comparison with the even values, s = 0 and 2.

Solitons in the Tonks–Girardeau gas with dipolar interactions

The existence of bright solitons in the model of the Tonks–Girardeau (TG) gas with dipole–dipole (DD) interactions is reported. The governing equation is taken as the quintic nonlinear Schrödinger equation (NLSE) with the nonlocal cubic term accounting for the DD attraction. In different regions of the parameter space (the dipole moment and atom number), matter-wave solitons feature flat-top or compacton-like shapes. For the flat-top states, the NLSE with the local cubic-quintic (CQ) nonlinearity is shown to be a good approximation. Specific dynamical effects are studied assuming that the strength of the DD interactions is ramped up or drops to zero. Generation of dark-soliton pairs in the gas shrinking under the action of the intensifying DD attraction is observed. Dark solitons exhibit particle-like collision behaviour. Peculiarities of dipole solitons in the TG gas are highlighted by comparison with the NLSE including the local CQ terms. Collisions between the bright solitons are also studied. In many cases, the collisions result in the merger of the solitons into a breather, due to a strong attraction between them.

Interactions between two-dimensional solitons in the diffractive–diffusive Ginzburg–Landau equation with the cubic–quintic nonlinearity

We report the results of systematic numerical analysis of collisions between two and three stable dissipative solitons in the two-dimensional (2D) complex Ginzburg–Landau equation (CGLE) with the cubic–quintic (CQ) combination of gain and loss terms. The equation may be realized as a model of a laser cavity which includes the spatial diffraction, together with the anomalous group-velocity dispersion (GVD) and spectral filtering acting in the temporal direction. Collisions between solitons are possible due to the Galilean invariance along the spatial axis. Outcomes of the collisions are identified by varying the GVD coefficient, β , and the collision “velocity” (actually, it is the spatial slope of the soliton’s trajectory). At small velocities, two or three in-phase solitons merge into a single standing one. At larger velocities, both in-phase soliton pairs and pairs of solitons with opposite signs suffer a transition into a delocalized chaotic state. At still larger velocities, all collisions become quasi-elastic. A new outcome is revealed by collisions between slow solitons with opposite signs: they self-trap into persistent wobbling dipoles, which are found in two modifications — horizontal at smaller β , and vertical if β is larger (the horizontal ones resemble “zigzag” bound states of two solitons known in the 1D CGL equation of the CQ type). Collisions between solitons with a finite mismatch between their trajectories are studied too.

Energy exchange between (3+1)D colliding spatiotemporal optical solitons in dispersive media with cubic—quintic nonlinearity

We investigate the energy exchange between (3+1)D colliding spatiotemporal solitons (STSs) in dispersive media with cubic-quintic (CQ) nonlinearity by numerical simulations. Energy exchange between two (3+1)D head on colliding STSs caused by their phase difference is observed, just as occurring in other optical media. Moreover, energy exchange between two head-on colliding STSs with different speeds is firstly shown in the CQ and saturable media. This phenomenon, we believe, may arouse some interest in the future studies of soliton collision in optical media.