Gaussian Solitons Research Articles

New Gaussons and traveling wave solutions for fractional nonlinear logarithmic models

The principal objective of this paper is to remark upon an investigation on four kinds of fractional Schrödinger equations. New logarithmic fractional models are presented for two of these equations. While the other equations are just considered in their fractional cases. All models are solved by utilizing the Adomain decomposition method. By this method some new Gaussian soliton wave solutions and traveling wave solutions are obtained.

Two-dimensional modulation instability and soliton clusters in nonlocal media with competing cubic–quintic nonlinearities

Using linear-stability analysis, two-dimensional modulation instability (MI) of plane wave is studied in nonlocal media with competing cubic–quintic (CQ) nonlinearities, which shows that MI can be effectively eliminated by strong nonlocality and competing quintic nonlinearity. Furthermore, propagation properties of higher-order soliton clusters, i.e., Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) solitons are also investigated. Bifurcated solutions of these solitons are obtained analytically with variational approach. We also demonstrate in detail the propagation dynamics of the HG and LG solitons with split-step Fourier transform.

Mode conversion of various solitons in parabolic and cross-phase potential wells

We numerically establish the controllable conversion between Laguerre–Gaussian and Hermite–Gaussian solitons in nonlinear media featuring parabolic and cross-phase potential wells. The parabolic potential maintains the stability of Laguerre–Gaussian and Hermite–Gaussian beams, while the actual conversion between the two modes is facilitated by the cross-phase potential, which induces an additional phase shift. By flexibly engineering the range of the cross-phase potential well, various higher-mode solitons can be generated at desired distances. Beams carrying orbital angular momentum can also be efficiently controlled by this method. In addition, other types of beams, such as sine complex-various-function Gaussian and hypergeometric-Gaussian vortex beams, can be periodically transformed and manipulated in a similar manner. Our approach allows the intricate internal relationships between different modes of beams to be conveniently revealed.

Dynamic properties, Gaussian soliton and chaotic behaviors of general Degasperis–Procesi model

Dynamic properties, Gaussian soliton and chaotic behaviors of general Degasperis–Procesi model

Twisted Gaussian Schell-model breathers and solitons in strongly nonlocal nonlinear media.

Based on the Snyder-Mitchell linear model and the cross-spectral density (CSD) function, the analytical propagation formula of twisted Gaussian Schell-model (TGSM) beams in strongly nonlocal nonlinear medium (SNNM) is derived. Then the propagation characteristics of TGSM beam are studied. It is found that the soliton radius is jointly determined by the initial power, coherence length, and twist factor; the degree of spatial coherence is adjusted by changing the twist factor without affecting the soliton intensity. In the case of non-soliton properties, there is a threshold of coherence length which makes partially coherent beams have the same evolution law as completely coherent beams. Furthermore, increasing the twist factor, decreasing the coherence length and initial power can improve the beam quality of the beam propagating in SNNM.

Variational quantum algorithm for Gaussian discrete solitons and their boson sampling

In the context of quantum information, highly nonlinear regimes, such as those supporting solitons, are marginally investigated. We miss general methods for quantum solitons, although they can act as entanglement generators or as self-organized quantum processors. We develop a computational approach that uses a neural network as a variational ansatz for quantum solitons in an array of waveguides. By training the resulting phase space quantum machine-slearning model, we find different soliton solutions, varying the number of particles and interaction strength. We consider Gaussian states that enable measuring the degree of entanglement and sampling the probability distribution of many-particle events. We also determine the probability of generating particle pairs and unveil that soliton bound states emit correlated pairs. These results may have a role in boson sampling with nonlinear systems and in quantum processors for entangled nonlinear waves.

Modulating chirped Gaussian vector solitons with group-velocity dispersion

We theoretically simulate the modulation of chirped Gaussian vector solitons with group-velocity dispersion at 1-μm. Several different kinds of soliton pulse parameters are considered for variation under the condition of same central wavelength (1064 nm) and slightly different central wavelengths (1063 nm and 1065 nm). The change of these parameters will influence the output properties of pulse shapes and corresponding optical spectra in orthogonal directions. For example, with same central wavelength, “3 + 3″ pseudo-higher-order vector solitons can be generated by changing time delay. And dual-wavelength vector solitons with multiple peaks/dips can be generated in many cases when central wavelengths are slightly different. Our simulations explore the modulation of vector solitons with group-velocity dispersion, which is conductive to the study of vector solitons that exist in long-distance nonlinear optical fiber system.

The interaction in nonlocal nonlinearity media under fractional effects

The propagation properties of the interaction of Airy Gaussian beams and solitons modeled by the fractional Schrödinger equation in nonlocal nonlinear media are demonstrated numerically. The transmission direction of beams is deflected in the nonlocal nonlinear media and the direction of deflection is related to the symbol of nonlocal nonlinear parameter. The interval parameter and Lévy index determine whether the beams form bound state or soliton in the transmission process. When the absolute value of the interval parameter and Lévy index are large, the beams cannot form solitons, but only a bound state beam. The transmission dynamics of the double Airy Gaussian beams and single soliton interaction are also discussed in this paper, as well as the numerical simulation results obtained for the double Airy Gaussian beams and double solitons interaction. These properties have potential application value for controlling the propagation and direction of light beams and have certain significance in the field of optical switches.

Generation, dynamics and bifurcation of high power soliton beams in cubic-quintic nonlocal nonlinear media

This article presents the generation and propagation dynamics of a high power Gaussian soliton beam through a highly nonlocal nonlinear media having cubic-quintic nonlinearity. Solitons are also generated with lesser explored Hermite super-Gaussian, Hermite cosh-Gaussian and Hermite cosh-super-Gaussian beam profiles. The governing nonlocal nonlinear Schrödinger equation yields matching solitons analytically using variational method as well as numerically using split-step Fourier method. Linear stability analysis identifies the parametric space for stability of the solitons against small perturbation. The variation of the system parameters leads to the bifurcation of the beam beyond a critical point. A parametric zone of bifurcation is identified. Some of the solitons are bistable too. The influence of quintic nonlinearity on generation, propagation and bifurcation is highlighted.

Topological properties and wave structures of Gilson–Pickering equation

The aim of this paper is to present the topological properties and wave structures of the Gilson–Pickering equation. By the traveling wave transformation, the corresponding traveling wave system of the original equation is obtained, and then a conserved quantity, namely the Hamiltonian is constructed via it. After that, the existences of the soliton and periodic solutions are established by the bifurcation method. To verify our conclusion explicitly, the corresponding exact traveling wave solutions are constructed. In particular, via the generalized trial equation method which is proposed in this paper, a special kind of soliton solution, namely the Gaussian soliton solution is given. To the best of our knowledge, this is the first time that a Gaussian soliton solution has been constructed to an equation with no logarithmic nonlinearity.

Stationary structures of spin-orbit coupled polariton condensates in Bessel lattices

<sec>Bessel optical lattice yields a non-spatially periodic column-symmetric optical lattice potential field, which has the characteristics of both infinite deep potential well and the ring-shaped potential well. A deep potential is formed in the center of the 0-order Bessel optical lattice. In the non-zero-order Beseel optical lattice, a ring-shaped shallow potential well with a central barrier can be formed. Exciton-polariton is a semi-light and semi-matter quasi-particle, which can achieve the Bose-Einstein condensate phase transition even at room temperature to form a polariton condensate. In addition, the polariton condensate is likely to realize sufficiently strong spin-orbit coupling due to the cavity-induced TE-TM splitting of the polariton energy levels. The polariton condensate can be realized at room temperature, and there can be spin-orbit coupling in it, which provides a new platform for the studying of quantum physics. </sec><sec>In this paper, the Bessel optical lattice is introduced into a polariton condensate. The stationary state structure of spinor two-component polariton condensate with spin-orbit coupling is investigated. By solving the Gross-Pitaevskii equation, we first give a stationary state structures of the polariton condensate both in the laboratory coordinate frame and in the rotating coordinate frame. Owing to the introduction of the Bessel optical lattice, the stationary state structures of polariton condensate are diverse. We dispaly the stationary state structures of the basic Gaussian solitons and multipole solitons in the central deep potential well in the laboratory coordinate frame, and the ring solitons and multipole solitons in the central shallow potential well. We also dispaly the vortex ring soliton that exists in the rotating coordinate frame, and the stationary state structure of the component separation caused by the spin-orbit interaction. We analyze not only the influences of the spin-orbit coupling on the stationary state structures in the two coordinate frames, but also the stability of the multipole solitons in the rotating coordinate frame. It is found that the multipole solitons formed in the ring-shaped shallow potential well have better stability than in the central deep potential well, and they can maintain the relative structure and spatial distribution for a long time in the rotation process. In the rotating coordinate frame, even if the two-component separation conditions are not satisfied, the introduction of spin-orbit coupling can cause the two components to separate.</sec>

Parabolic Frequency on Manifolds

Abstract We prove monotonicity of a parabolic frequency on static and evolving manifolds without any curvature or other assumptions. These are parabolic analogs of Almgren’s frequency function. When the static manifold is Euclidean space and the drift operator is the Ornstein–Uhlenbeck operator, this can been seen to imply Poon’s frequency monotonicity for the ordinary heat equation. When the manifold is self-similarly evolving by the Ricci flow, we prove a parabolic frequency monotonicity for solutions of the heat equation. For the self-similarly evolving Gaussian soliton, this gives directly Poon’s monotonicity. Monotonicity of frequency is a parabolic analog of the 19th century Hadamard three-circle theorem about log convexity of holomorphic functions on C. From the monotonicity, we get parabolic unique continuation and backward uniqueness.

Accessible solitons in three-dimensional parabolic cylindrical coordinates

A class of self-similar beams, named three-dimensional (3D) spatiotemporal parabolic accessible solitons, are introduced in the 3D highly nonlocal nonlinear media. We obtain exact solutions of the 3D spatiotemporal linear Schrödinger equation in parabolic cylindrical coordinates by using the method of separation of variables. The 3D localized structures are constructed with the help of the confluent hypergeometric Tricomi functions and the Hermite polynomials. Based on such an exact solution, we graphically display three different types of 3D beams: the Gaussian solitons, the ring necklace solitons, and the parabolic solitons, by choosing different mode parameters. We also perform direct numerical simulation to discuss the stability of local solutions. The procedure we follow provides a new method for the manipulation of spatiotemporal solitons.

Gaussian soliton pairs in an unbiased photorefractive crystal due to the pyroelectric effect

The characteristics of the dynamical evolution of incoherently coupled Gaussian soliton pairs in unbiased photorefractive media forming solely due to the pyroelectric effect are studied for the first time. A parameter space spawning a large set of bright Gaussian soliton pairs in unbiased photorefractive crystals having a large pyroelectric coefficient has been found. These soliton pairs can exist if the light beams have similar polarization, equal wavelength and are mutually incoherent. A change in the temperature of the photorefractive crystal induces a temporary pyroelectric field resulting in the formation of a space-charge field. The existence curve for these soliton pairs is studied for different values of the temperature change. It is also established that both individual components of the soliton pair should have equal spatial widths for stable propagation. The whole parameter space in which these soliton pairs exist as a changeless entity has been investigated in terms of the spatial width of the soliton pair, their peak power, and the magnitude of temperature change of the crystal. The magnitude of the temperature change profoundly affects the evolution characteristics of these Gaussian soliton pairs. The stability of such soliton pairs has been demonstrated using the paraxial theory. Bistable states have been shown to exist in the system irrespective of the magnitude of the temperature change for the case when both components of the soliton pair have equal power. A lithium niobate (LN) crystal has been considered to illustrate the relevant results.

Coherent superposition propagation of Laguerre–Gaussian and Hermite–Gaussian solitons

In this work, we mainly investigate the propagation of the superimposed field of Laguerre–Gaussian and Hermite–Gaussian solitons in nonlocal nonlinear Schrödinger equation. The propagation expression is obtained and the propagation properties are analyzed. The results show that the propagation properties of the superimposed field are closely related to the input power and the characteristic parameters. For different input powers, the superimposed field will show periodic changes of broadening or compression while propagating. Different characteristic parameters determine which soliton occupies the dominant position in propagation.

Gaussian-like and flat-top solitons of atoms with spatially modulated repulsive interactions

Solitons, nonlinear particle-like excitations with inalterable properties (amplitude, shape, and velocity) as they propagate, are omnipresent in many branches of science---and in physics in particular. Flat-top solitons are a novel type of bright solitons that have not been well explored in pure nonlinear media. Here, a model of nonlinear Kerr (cubic) media of ultracold atoms with spatially modulated repulsive interactions is proposed and shown to support a vast variety of stable flat-top matter-wave solitons, including one-dimensional (1D) flat-top fundamental and multipole solitons, two-dimensional (2D) flat-top fundamental and vortex solitons. We demonstrate that by varying the relevant physical parameters (nonlinearity coefficient and chemical potential) the ordinary bright (gaussian) solitons can transform into the novel flat-top solitons. The (in-)stability domains of the flat-top soliton families are checked by means of linear stability analysis and reconfirmed by direct numerical simulations. This model is generic in the contexts of nonlinear optics and Bose-Einstein condensates, which provide direct experimental access to observe the predicted solutions.

Manipulation of Gaussian derivative pulses and vector solitons in fiber lasers with analytic modeling by the coupled complex Ginzburg–Landau equations

Both positive- and negative-polarity Gaussian monocycle and doublet pulses, for which the pulse shapes are the first and second derivatives of Gaussian functions plus some continuous-wave backgrounds respectively, are generated in a ring-cavity erbium-doped fiber laser from polarization-locked vector solitons by using passive optical technology. The pulse states are switchable and are found to be the superposition of bright and dark solitons with different widths, amplitudes, time delays, polarizations, and wavelengths. Qualitative analysis of the properties of vector solitons are performed by solving coupled complex Ginzburg–Landau equations, and analytic modeling of the fiber laser is presented. By representing the envelopes of bright soliton by sech- and dark soliton by tanh-based functions, the incoherent superposition of these two soliton components have simulated the experimental observations, and the underlying mechanisms for the formation of monocycle and doublet pulses are attributed to the polarization locking of bright and dark solitons. The theoretical modeling is used to calculate the pulse parameters of the fiber laser, and the intracavity birefringence can be estimated. The results of tunable optical vector solitons are compared with atomic solitons in the system of Bose–Einstein condensation. Since the governing equations for soliton generation in fiber lasers and Bose–Einstein condensation have many properties in common, thus the simulation and propagation of pulsating waves may open a new route to explore the classical solitary dynamics in nonlinear optics and its quantum analogy in ultracold fields.

Soliton guidance and nonlinear coupling for polarized vector spiraling elliptic Hermite–Gaussian beams in nonlocal nonlinear media**Project supported by the National Natural Science Foundation of China (Grant No. 11604199) and the China Scholarship Council (Grant No. 201708410236).

We investigate the incoherent beams with two orthogonal polarizations in nonlocal nonlinear media, one of which is a fundamental Gaussian beam and the other is spiraling elliptic Hermite–Gaussian beam carrying the orbital angular momentum (OAM). Using the variational approach, we obtain the critical power and the critical OAM required for the vector spiraling elliptic Hermite–Gaussian solitons. In the strong nonlocality region, two components of the vector beam contribute to the nonlinear refractive index in a linear manner by the sum of their respective power. The nonlinear refractive index exhibits a circularly symmetrical profile in despite of the elliptic shapes for spiraling Hermite–Gaussian beams. We find that in the strong nonlocality region, the critical power and the rotational velocity are the same regardless of the relative ratio of the constituent powers. The nonlinear refractive index loses its circular symmetry in weak nonlocality region, and the nonlinear coupling effect is observed. Due to the radiation of the OAM, the damping of the rotation is predicted, and can be suppressed by decreasing the proportion of the spiraling elliptic component of the vector beam.

Chirped optical Gausson perturbation with quadratic–cubic nonlinearity by collective variables

Here, collective variables approach is applied to obtain optical solitons in the presence of perturbation terms. Gaussian solitons are selected to retain the pulse to establish soliton setting. The numerical simulations of soliton parameters are performed for specific values of the Gaussian pulse parameters.

Stability analysis, solition solutions and Gaussian solitons of the generalized nonlinear Schrödinger equation with higher order terms

Purpose The purpose of this paper is to study stability analysis, solition solutions and Gaussian solitons of the generalized nonlinear Schrödinger equation with higher order terms, which can be used to describe the propagation properties of optical soliton solutions. Design/methodology/approach The authors apply the ansatz method and the Hamiltonian system technique to find its bright, dark and Gaussian wave solitons and analyze its modulation instability analysis and stability analysis solution. Findings The results imply that the generalized nonlinear Schrödinger equation has bright, dark and Gaussian wave solitons. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior. Some constraint conditions are provided which can guarantee the existence of solitons. The authors analyze its modulation instability analysis and stability analysis solution. Originality/value These results may help us to further study the local structure and the interaction of solutions in generalized nonlinear Schrödinger -type equations. The authors hope that the results provided in this work can help enrich the dynamic behavior of the generalized nonlinear Schrödinger--type equations.