Fractional Solitons Research Articles

Vector multipole solitons of fractional-order coupled saturable nonlinear Schrödinger equation

Three kinds of vector multipole solitons of fractional coupled saturable nonlinear Schrödinger equation are reported, including fractional dipole-dipole, dipole-tripole and tripole-dipole vector soliton solutions. Firstly, their existence domains, which are modulated by potential function parameters, are constructed in a certain interval. Secondly, the stable regions of three kinds of vector multipole solitons, which are modulated by the soliton power of each component, are found. The properties of solitons are explored through these existence and stability domains. Finally, the stability of three kinds of fractional vector multipole solitons is verified by the numerical evolution. Compared with the integer-order results, there are differences in the existence and stable regions of soliton solutions, and the Lévy index affects the existence and stability of vector multipole solitons.

Modulation instability and extraction of fractional optical solitons in the presence of generalized Kudryashov’s law and dual form of non-local nonlinearity

This research delves into the study of optical solitons governed by Kudryashov’s law of nonlinear refractive index, incorporating a conformable fractional derivative, perturbed terms, and a quadrupled-power law in optical fibers. Employing a novel approach known as the generalized exp(− S(ζ) ) expansion method, we identify novel optical soliton solutions characterized by exponential, rational, and hyperbolic functions featuring a conformable fractional derivative. These solutions manifest as unified bright-dark, singular, singular periodic, kink, anti-kink, and novel solitary waves. We employ 3D, contour, and 2D plots to analyze the behavior of these solutions across different temporal and fractional-order derivative parameters, offering an understanding of their physical interpretations. Furthermore, we perform modulation instability (MI) analysis that depends on standard linear stability analysis to derive the MI gain spectrum, demonstrating how well our methodology works for nonlinear problems in the natural sciences and engineering domains. This approach provides an effective way to investigate optical solutions in various integer and fractional Schrödinger equations.

Interactions between fractional solitons in bimodal fiber cavities

AbstractWe introduce a system of fractional nonlinear Schrödinger equations (FNLSEs) which model the copropagation of optical waves carried by different wavelengths or mutually orthogonal circular polarizations in fiber‐laser cavities with the effective fractional group‐velocity dispersion (FGVD), which were recently made available to the experiment. In the FNLSE system, the FGVD terms are represented by the Riesz derivative, with the respective Lévy index (LI). The FNLSEs, which include the self‐phase‐modulation (SPM) nonlinearity, are coupled by the cross‐phase‐modulation (XPM) terms, and separated by a group‐velocity (GV) mismatch (rapidity). By means of systematic simulations, we analyze collisions and bound states of solitons in the XPM‐coupled system, varying the LI and GV mismatch. Outcomes of collisions between the solitons include rebound, conversion of the colliding single‐component solitons into a pair of two‐component ones, merger of the solitons into a breather, their mutual passage leading to excitation of intrinsic vibrations, and the elastic interaction. Families of stable two‐component soliton bound states are constructed too, featuring a rapidity which is intermediate between those of the two components.

Formation and propagation of fundamental and vortex soliton families in 1D and 2D dual-Lévy-index fractional nonlinear Schrödinger equations with cubic–quintic nonlinearity

We address effects of interplay of two different fractional-diffraction terms, corresponding to different Lé vy indices (LIs), in the framework of nonlinear Schrödinger equation (NLSE) with cubic–quintic nonlinearity (dual-LI fractional NLSE ), in one- and two-dimensional (1D and 2D) settings. The critical (in 1D) and supercritical (in 2D) wave collapses are suppressed in the presence of a defocusing quintic term, making it possible to produce stable localized modes, including fundamental and vortical ones. In 2D, families of fundamental and vortex solitons, with topological charges S=0,1,2,3, are produced in a numerical form and by dint of the variational approximation (VA). In particular, the threshold power necessary for the existence of 2D fundamental solitons is predicted by the VA, being close to the numerical results. In the case of fractional-diffraction terms with unequal LIs acting in two transverse directions, anisotropic 2D fundamental and vortex solitons are constructed. Stability of the solitons is investigated for small perturbations governed by the linearized equations, and results are corroborated by direct simulations of the perturbed evolution. Those vortex solitons which are unstable are split by azimuthal perturbations into fragments, whose number is determined by the shape of the perturbation eigenmode with the largest growth rate. These results extend the concept of the 2D fractional diffraction and solitons to media with the anisotropic fractionality.

Basic fractional nonlinear-wave models and solitons

This review article provides a concise summary of one- and two-dimensional models for the propagation of linear and nonlinear waves in fractional media. The basic models, which originate from Laskin’s fractional quantum mechanics and more experimentally relevant setups emulating fractional diffraction in optics, are based on the Riesz definition of fractional derivatives, which are characterized by the respective Lévy indices. Basic species of one-dimensional solitons, produced by the fractional models which include cubic or quadratic nonlinear terms, are outlined too. In particular, it is demonstrated that the variational approximation is relevant in many cases. A summary of the recently demonstrated experimental realization of the fractional group-velocity dispersion in fiber lasers is also presented.

Analytical soliton solutions for the beta fractional derivative Gross–Pitaevskii system with linear magnetic and time dependent laser interactions

The local conformable beta Atangana derivative will be considered for the introduction of the fractional Gross–Pitaevskii model with conformable derivatives of beta type. Analytical expressions for soliton solutions are constructed by sub-equation method with elliptical functions. The main goal of the current research is to determine the general behavior of the soliton solutions, their dependence on the elliptical parameter and the influence of the fractional order parameter on the time and space scales of the solutions. New entire family of solitons were determined by considering the arising constrains over the parameters of the nonlinear fractional Gross–Pitaevskii system. The analytical expressions for the soliton solutions constructed for the fractional order case reduce to the well known solitons previously reported for hyperbolic and periodic tan-type singular solutions for the integer order limit value, when special cases of the Jacobi elliptic functions are considered. Solitons properties are depicted in 3-D level and 2-D illustrations. The fractional solitons here introduced possess some interesting time evolution behavior observed in the 3-D representations, these time properties are not present in the integer order case and has an important dependency on the fractional parameter of the beta derivative. The solitons here introduced for the nonlinear fractional Gross–Pitaevskii equation will be very useful in future works where additional interactions will be introduced for the study of different Bose–Einstein condensation phenomena, the coupled quasi-one dimensional Gross- Pitaevskii equation or other nonlinear phenomena where non regular oscillations will be involved.

Kink phenomena of the time-space fractional Oskolkov equation

<p>In this study, we applied the Riccati-Bernoulli sub-ODE method and Bäcklund transformation to analyze the time-space fractional Oskolkov equation for kink solutions by matching the coefficients and optimal series parameters. The time-space fractional Oskolkov equation is used to analyze the behavior of solitons for different applications such as fluid dynamics and viscoelastic flow. The kink solutions derived have important consequences for stability analysis and interaction dynamic in these systems, and these are useful in controlling the physical behaviour of systems described by this equation. Such effects are illustrated by 2D and 3D plots, showing that the proposed model can handle both fractional and integer-order solitons with different but equally efficient outcomes. This research contributes to a valuable analytical method that can determine and manage processes in diversified systems based on fractional differential equations. This work provides a basis for subsequent analysis in other branches of science and technology in which the fractional Oskolkov model is used.</p>

Fractional topological solitons in nonlinear viscoelastic ribbons with tunable speed

A unique type of domain-wall solitons, fractional topological solitons, have been recently theoretically shown to arise in minimal surface ribbons and demonstrate topological robustness. Here we study these fractional topological solitons experimentally using 3D printed helicoid ribbons and computationally using finite-element analysis (FEA). We show that these fractional solitons propagate at constant speed despite the strong dissipation in the viscoelastic material the ribbon consists of, and this speed can be sensitively controlled by an axial load. The nonlinear viscoelastic FEA quantitatively captures fine features of this fractional soliton. The results verify the predicted fractional solitons, and more importantly, open a new pathway of realizing and controlling topological fractional excitations in systems with complex material features including nonlinearity and viscoelasticity.

Fractional solitons: New phenomena and exact solutions

The fractional solitons have demonstrated many new phenomena, which cannot be explained by the traditional solitary wave theory. This paper studies some famous fractional wave equations including the fractional KdV–Burgers equation and the fractional approximate long water wave equation by a modified tanh-function method. The solving process is given in details, and new solitons can be rigorously explained by the obtained exact solutions. This paper offers a new window for studying fractional solitons.

The Sensitive Visualization and Generalized Fractional Solitons’ Construction for Regularized Long-Wave Governing Model

The solution of partial differential equations has generally been one of the most-vital mathematical tools for describing physical phenomena in the different scientific disciplines. The previous studies performed with the classical derivative on this model cannot express the propagating behavior at heavy infinite tails. In order to address this problem, this study addressed the fractional regularized long-wave Burgers problem by using two different fractional operators, Beta and M-truncated, which are capable of predicting the behavior where the classical derivative is unable to show dynamical characteristics. This fractional equation is first transformed into an ordinary differential equation using the fractional traveling wave transformation. A new auxiliary equation approach was employed in order to discover new soliton solutions. As a result, bright, periodic, singular, mixed periodic, rational, combined dark–bright, and dark soliton solutions were found based on the constraint relation imposed on the auxiliary equation parameters. The graphical visualization of the obtained results is displayed by taking the suitable parametric values and predicting that the fractional order parameter is responsible for controlling the behavior of propagating solitary waves and also providing the comparison between fractional operators and the classical derivative. We are confident about the vital applications of this study in many scientific fields.

Nondegenerate solitons in the integrable fractional coupled Hirota equation

In this paper, based on the nonlinear fractional equations proposed by Ablowitz, Been, and Carr in the sense of Riesz fractional derivative, we explore the fractional coupled Hirota equation and give its explicit form. Unlike the previous nonlinear fractional equations, this type of nonlinear fractional equation is integrable. Therefore, we obtain the fractional n-soliton solutions of the fractional coupled Hirota equation by inverse scattering transformation in the reflectionless case. In particular, we analyze the one- and two-soliton solutions and prove that the fractional two-soliton can also be regarded as a linear superposition of two fractional single solitons as |t|→∞. Moreover, we obtain the nondegenerate fractional soliton solutions and give a simple analysis for them.

Sensitive Demonstration of the Twin-Core Couplers including Kerr Law Non-Linearity via Beta Derivative Evolution

To obtain new solitary wave solutions for non-linear directional couplers using optical meta-materials, a new extended direct algebraic technique (EDAT) is used. This model investigates solitary wave propagation inside a fiber. As a result, twin couplers are the subject of this study. Kerr law is the sort of non-linearity addressed there. Because it offers solutions to problems with large tails or infinite fluctuations, the resulting solution set is more generalized than the current solution because it is turned into a fractional-order derivative. Furthermore, the found solutions are fractional solitons with spatial–temporal fractional beta derivative evolution. In intensity-dependent switches, these nonlinear directional couplers also serve as limiters. Non-linearity alters the transmission constants of a system’s modes. The significance of the beta derivative parameter and mathematical approach is demonstrated graphically, with a few of the extracted solutions. A parametric analysis revealed that the fractional beta derivative parameter has a significant impact on the soliton amplitudes. With the aid of the advanced software tools for numerical computations, the categories of semi-dark solitons, singular dark-pitch solitons, single solitons of Type-1 along with 2, intermingled hyperbolically, trigonometric, and rational solitons were established and evaluated. We also discussed sensitivity analysis, which is an inquiry that determines how sensitive our system is. A comparative investigation via different fractional derivatives was also studied in this paper so that one can easily understand the correlation with other fractional derivatives. The findings demonstrate that the approach is simple and efficient and that it yields generalized analytical results. The findings will be extremely beneficial in examining and comprehending physical issues in nonlinear optics, specifically in twin-core couplers with optical metamaterials.

Dynamical behaviours and fractional alphabetical-exotic solitons in a coupled nonlinear electrical transmission lattice including wave obliqueness

Dynamical behaviours and fractional alphabetical-exotic solitons in a coupled nonlinear electrical transmission lattice including wave obliqueness

Optical solitons in the generalized space–time fractional cubic-quintic nonlinear Schrödinger equation with a [formula omitted]-symmetric potential

In the presence of complexified parity reflection–time reversal (PT)−symmetric Scarff-II potential, we consider the generalized space–time fractional cubic-quintic nonlinear Schrödinger (FCQNLS) equation. By implementing fractal derivative variable transformation, we derive the fractional optical soliton solutions for the considered model. We consider two distinct forms of cubic-quintic nonlinearities, such as (i) focusing cubic-focusing quintic and (ii) defocusing cubic-focusing quintic for deriving the fractional optical soliton solutions. We investigate how modifying the temporal and space fractional-order parameters affects these fractional solitons. Our observations reveal that the rise in fractional-order parameters leads to the desired soliton profiles. We also examine further the effects of potential strengths on the fractional soliton profiles that are obtained.

Soliton and rogue wave solutions of the space–time fractional nonlinear Schrödinger equation with [formula omitted]-symmetric and time-dependent potentials

We derive fractional soliton and rogue wave solutions of the space–time fractional nonlinear Schrödinger (FNLS) equation in the existence of complex parity reflection–time reversal (PT)−symmetric and time-dependent potentials. We find that the fractional derivative variable transformation is a good approximation to reduce the space–time FNLS equation into its conventional counterpart. Utilizing the localized solutions of the reduced conventional system, we construct the fractional localized solutions for the system under consideration. We study the dynamics of localized fractional solitons with PT-symmetric Rosen–Morse, Scarff-II and time dependent potentials. We explore the impact of dynamical properties on the constructed fractional solitons by changing time, space fractional-order parameter, real and imaginary components of potential strengths. Our observations reveal that the soliton and RW profiles get distorted for low values of time and space fractional-order parameters, and they show the usual features when these parameters are close to unity. Furthermore, we also report that the intensity of localized profiles increases by varying the strength of the real part of potentials, and the unstable behaviour is exhibited for higher strengths of the imaginary part of potentials.

Dissipative solitons of the nonlinear fractional Schrödinger equation with PT-symmetric potential

We study the fractional dissipative solitons with nonlinear loss and linear gain supported by a special class of PT-symmetric potential in focused nonlinear Kerr medium. The effective width, existence and stability of fractional solitons are discussed. In such a potential, the stability of solitons changes at lower energy. Once the stability of symmetric solitons is destroyed, asymmetric solitons and symmetric solitons coexist. With the increase of linear gain coefficient, the effective width decreases first and then increases. With the increase of Lévy index, the amplitude of solitons decreases. In addition, the change of nonlinear loss coefficient has almost no effect on the soliton contour.

Creation and annihilation of mobile fractional solitons in atomic chains

Localized modes in one-dimensional (1D) topological systems, such as Majonara modes in topological superconductors, are promising candidates for robust information processing. While theory predicts mobile integer and fractional topological solitons in 1D topological insulators, experiments so far have unveiled immobile, integer solitons only. Here we observe fractionalized phase defects moving along trimer silicon atomic chains formed along step edges of a vicinal silicon surface. By means of tunnelling microscopy, we identify local defects with phase shifts of 2π/3 and 4π/3 with their electronic states within the band gap and with their motions activated above 100 K. Theoretical calculations reveal the topological soliton origin of the phase defects with fractional charges of ±2e/3 and ±4e/3. Additionally, we create and annihilate individual solitons at desired locations by current pulses from the probe tip. Mobile and manipulable topological solitons may serve as robust, topologically protected information carriers in future information technology.

Fractional Excitations in Non-Euclidean Elastic Plates

We show that minimal-surface non-Euclidean elastic plates share the same low-energy effective theory as Haldane's dimerized quantum spin chain. As a result, such elastic plates support fractional excitations, which take the form of charge-1/2 solitons between degenerate states of the plate, in strong analogy to their quantum counterpart. These fractional excitations exhibit properties similar to fractional excitations in quantum fractional topological states and in Haldane's dimerized quantum spin chain, including deconfinement and braiding, as well as unique new features such as holographic properties and diodelike nonlinear response, demonstrating great potentials for applications as mechanical metamaterials.

The fractional comparative study of the non-linear directional couplers in non-linear optics

A new extended direct algebraic method is applied to extract new traveling wave solutions for non-linear directional couplers with the optical meta-materials. This model studies light distribution from the main fiber into other branch fibers, which can be one or more than one. So, for this reason, twin couplers and multiple core couplers are studied in this paper. Further two cases in multiple core couplers namely multiple core couplers near the neighbor and multiple core couplers with a complete neighbor are exhaustively discussed. The non-linearity type that is considered here is Kerr Law. The obtained solutions set is more generalized than the existing solution as it is transformed into fractional-order derivative because it gives the solutions of problems with heavy tails or infinity fluctuations. Furthermore, the obtained solutions contain almost all types of solitons such as dark, bright, dark-bright, rational, periodic, breather, and singular soliton solutions, etc. Moreover, the obtained solutions are fractional solitons with Atangana-Baleanu and modified Riemann–Liouville fractional operator. These non-linear directional couplers also act as limiters in intensity-dependent switches. The propagation constants of the modes of a system are changed by non-linearity. In order to discuss the physical nature of couplers, 2D and 3D figures are also shown.

Quasi-stable fractional vortex solitons in nonlocal nonlinear media

This paper studied the propagations of fractional vortex beams in nonlocal nonlinear media, and found that quasi-stable solitons can form when the nonlocality is strong enough and the initial power is equal to the critical power. The propagation properties, including intensity patterns, phase structures, transverse energy flows, and rotation period, were all investigated. The critical power of solitons with different topological charges is the same, however, the orbital angular momentum (OAM) of that increases with the increase of fractional topological charge. In addition, we revealed that fractional vortex solitons have lateral shifts in the y-direction, and the shift values can be controlled by the fractional topological charge. Obviously, the quasi-stable fractional vortex solitons have important academic value and potential application to optical switches, optical wrenches and optical communications, etc.