Accessible solitary wave families of the generalized nonlocal nonlinear Schrödinger equation

Two-dimensional accessible solitary wave families of the generalized nonlocal nonlinear Schr?dinger equation are obtained by utilizing superpositions of various single accessible solitary solutions. Specific values of soliton parameters are selected as initial conditions and the superposition of known single solitary solutions in the highly nonlocal regime are launched into the nonlocal nonlinear medium with a Gaussian response function, to obtain novel numerical solitary solutions of improved stability. Our results reveal that in nonlocal media with the Gaussian response the higher-order spatial accessible solitary families can exist in various forms, such as asymmetric necklace, asymmetric fractional, and symmetric multipolar necklace solitons.     

Fractional optical solitons

This Letter shows that soliton propagation can be described by an extended NLS equation which incorporates fractional dispersion and a fractional nonlinearity. The fractional dispersive term is written in terms of Grünwald-Letnikov fractional derivatives (FDs). Forward and backward FDs are introduced in order to satisfy the conservation of energy. It is found that the soliton solutions of this model form a continuous family and are stable. The Vakhitov-Kolokolov criterion is used to confirm the stability of these fractional solitons.     

Three-dimensional spatiotemporal solitary waves in strongly nonlocal media

We construct a class of three-dimensional strongly nonlocal spatiotemporal solitary waves of the nonlocal nonlinear Schrödinger equation, by using superpositions of single accessible solitons as initial conditions. Evolution of such solitary waves, termed the accessible light bullets, is studied numerically by choosing specific values of topological charges and other solitonic parameters. Our numerical results reveal that in strongly nonlocal nonlinear media with a Gaussian response function, different classes of accessible spatiotemporal solitons can be generated and controlled by tailoring different soliton parameters.     

Ince-Gaussian solitons in strongly nonlocal nonlinear media

We have introduced a novel class of higher-order spatial optical Ince-Gaussian solitons (IGSs) that constitute the third complete family of exact and orthogonal soliton solutions of the Snyder-Mitchell model. The transverse structure of the IGSs is characterized by the Ince polynomials and has an inherent elliptical symmetry. The IGSs form the exact and continuous transition modes between Hermite-Gaussian solitons and Laguerre-Gaussian solitons.     

Analytical treatments of the space–time fractional coupled nonlinear Schrödinger equations

Under investigation in this work is a (\(2+1\))-dimensional the space–time fractional coupled nonlinear Schrödinger equations, which describes the amplitudes of circularly-polarized waves in a nonlinear optical fiber. With the aid of conformable fractional derivative and the fractional wave transformation, we derive the analytical soliton solutions in the form of rational soliton, periodic soliton, hyperbolic soliton solutions by four integration method, namely, the extended trial equation method, the \(\exp (-\,\Omega (\eta ))\)-expansion method and the improved \(\tan (\phi (\eta )/2)\)-expansion method and semi-inverse variational principle method. Based on the the extended trial equation method, we derive the several types of solutions including singular, kink-singular, bright, solitary wave, compacton and elliptic function solutions. Under certain condition, the 1-soliton, bright, singular solutions are driven by semi-inverse variational principle method. Based on the analytical methods, we find that the solutions give birth to the dark solitons, the bright solitons, combine dark-singular, kink, kink-singular solutions with fractional order for nonlinear fractional partial differential equations arise in nonlinear optics.     

Symmetric and antisymmetric vector solitons for the fractional quadric-cubic coupled nonlinear Schrödinger equation

The fractional quadric-cubic coupled nonlinear Schrödinger equation is concerned, and vector symmetric and antisymmetric soliton solutions are obtained by the square operator method. The relationship between the Lévy index and the amplitudes of vector symmetric and antisymmetric solitons is investigated. Two components of vector symmetric and antisymmetric solitons show a positive and negative trend with the Lévy index, respectively. The stability intervals of these solitons and the propagation constants corresponding to the maximum and minimum instability growth rates are studied. Results indicate that vector symmetric solitons are more stable and have better interference resistance than vector antisymmetric solitons.     

拉曼增益对双折射光纤中孤子传输特性的影响

本文采用考虑拉曼增益的耦合非线性薛定谔方程,利用分步傅里叶方法求解并仿真模拟了光孤子脉冲在不同性质的双折射光纤中传输时的演化过程.结果表明,拉曼增益可以有效抑制非线性耦合导致的孤子漂移,同时会导致光孤子脉冲峰值在传输时不断增大,产生拉曼放大效应.拉曼增益也可以有效抑制双折射光纤中传输的相邻光孤子之间的相互作用.     

1D Solitons in Saturable Nonlinear Media with Space Fractional Derivatives

Two decades ago, standard quantum mechanics entered into a new territory called space-fractional quantum mechanics, in which wave dynamics and effects are described by the fractional Schrödinger equation. Such territory is now a key and hot topic in diverse branches of physics, particularly in optics driven by the recent theoretical proposal for emulating the fractional Schrödinger equation. However, the light-wave propagation in saturable nonlinear media with space fractional derivatives is yet to be clearly disclosed. Here, such nonlinear optics phenomenon is theoretically investigated based on the nonlinear fractional Schrödinger equation with nonlinear lattices—periodic distributions of either focusing cubic (Kerr) or quintic saturable nonlinearities—and the existence and evolution of localized wave structures allowed by the model are addressed. The model upholds two kinds of one-dimensional soliton families, including fundamental solitons (single peak) and higher-order solitonic structures consisting of two-hump solitons (in-phase) and dipole ones (anti-phase). Notably, the dipole solitons can be robust stable physical objects localized merely within a single well of the nonlinear lattices—previously thought impossible. Linear-stability analysis and direct simulations are executed for both soliton families, and their stability regions are acquired. The predicted solutions can be readily observed in optical experiments and beyond.     

Painlevé Analysis,Soliton Collision and Bäcklund Transformation for the (3+1)-Dimensional Variable-Coefficient Kadomtsev-Petviashvili Equation in Fluids or Plasmas

In this paper, we investigate a (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation, which can describe the nonlinear phenomena in fluids or plasmas. Painlevé analysis is performed for us to study the integrability, and we find that the equation is not completely integrable. By virtue of the binary Bell polynomials, bilinear form and soliton solutions are obtained, and Bäcklund transformation in the binary-Bell-polynomial form and bilinear form are derived. Soliton collisions are graphically discussed: the solitons keep their original shapes unchanged after the collision except for the phase shifts. Variable coefficients are seen to affect the motion of solitons: when the variable coefficients are chosen as the constants, solitons keep their directions unchanged during the collision; with the variable coefficients as the functions of the temporal coordinate, the one soliton changes its direction.     

Symmetric and Anti-Symmetric Solitons of the Fractional Second-and Third-Order Nonlinear Schr?dinger Equation

The fractional second-and third-order nonlinear Schr?dinger equation is studied,symmetric and antisymmetric soliton solutions are derived,and the influence of the Levy index on different solitons is analyzed.The stability and stability interval of solitons are discussed.The anti-interference ability of stable solitons to the small disturbance shows a good robustness.     

Fraction‐Dimensional Accessible Solitons in a Parity‐Time Symmetric Potential

By using the modified Snyder‐Mitchell (MSM) model, which can describe the propagation of a paraxial beam in fractional dimensions (FDs), we find the exact "accessible soliton” solutions in the strongly nonlocal nonlinear media with a self‐consistent parity‐time (PT) symmetric complex potential. The exact solutions are constructed with the help of two special functions: the complex Gegenbauer and the generalized Laguerre polynomials in polar coordinates, parametrized by two nonnegative integer indices ‐ the radial and azimuthal mode numbers (n,m), and the beam modulation depth. By the choice of different soliton parameters, the intensity and angular profiles display symmetric and asymmetric structures. We believe that it is important to explore the MSM model in FDs and PT‐symmetric potentials, for a better understanding of nonlinear FD physical phenomena. Different physical systems in which the model might be of relevance are briefly discussed.     

Double Loops and Pitchfork Symmetry Breaking Bifurcations of Optical Solitons in Nonlinear Fractional Schrödinger Equation with Competing Cubic-Quintic Nonlinearities

Symmetry breaking bifurcations of solitons are investigated in framework of a nonlinear fractional Schrödinger equation (NLFSE) with competing cubic-quintic nonlinearity. Some prototypical characteristics of the symmetry breaking, featured by transformations of symmetric and antisymmetric soliton families into asymmetric ones, are found. Stable asymmetric solitons emerge from unstable symmetric and antisymmetric ones by way of two different symmetry breaking scenarios. A twisting branch, featured with double loops bifurcation, bifurcates off from the base branch of symmetric soliton solutions and crosses it, then merges into the base branch driven by the competitive nonlinear effect. A supercritical pitchfork bifurcation is bifurcated from the branch of antisymmetric soliton solutions and gives rise to a supercritical pitchfork bifurcation. Stability of the soliton families is explored by linear stability analysis. With the increase of the Lévy index, stability region induced by the twisting loops bifurcation is expanded. However, stability region of the pitchfork bifurcation is shrunk on the parameter plane of the Lévy index and the soliton power.     

Symbolic computation on the bright soliton solutions for the generalized coupled nonlinear Schrödinger equations with cubic–quintic nonlinearity

Under investigation in this paper are the generalized coupled nonlinear Schrödinger equations with cubic–quintic nonlinearity which describe the effects of the quintic nonlinearity on the ultrashort optical soliton pulse propagation in the non-Kerr media. Via the dependent variable transformation and Hirota method, the bilinear form is derived. Based on the bilinear form obtained, the one-, two- and three-soliton solutions are presented in the form of exponential polynomials with the help of symbolic computation. Propagation and interactions of solitons are investigated analytically and graphically. Evolution of one soliton is discussed with the analysis of such physical quantities as the soliton amplitude, width, velocity, initial phase and energy. Interactions of the solitons appear in the forms of the repulsion or attraction alternately and propagation in parallel. Inelastic and head-on interactions of the solitons are also showed. Finally, via the asymptotic analysis, conditions of the elastic and inelastic interactions are obtained.     

Optical soliton perturbation for time fractional resonant nonlinear Schrödinger equation with competing weakly nonlocal and full nonlinearity

In this article, we retrieve optical soliton solutions of the perturbed time fractional resonant nonlinear Schrödinger equation having competing weakly nonlocal and full nonlinearity. We study the equation for two different forms of nonlinearity, namely Kerr law and anti-cubic law. The F-expansion method along with fractional complex transformation is used to obtain the optical solitons. Moreover, the existence of these solitons are guaranteed with the constraint relations between the model coefficients and the traveling wave frequency coefficient.     

Soliton Solutions,B/icklund Transformations and Lax Pair for a （3 ＋ 1）-Dimensional Variable-Coefficient Kadomtsev-Petviashvili Equation in Fluids

Under investigation in this paper is a （3 q- 1）-dimensional variable-coefficient Kadomtsev-Petviashvili equation, which describes the propagation of surface and internal water waves. By virtue of the binary Bell polynomials, symbolic computation and auxiliary independent variable, the bilinear forms, soliton solutions, Backlund transformations and Lax pair are obtained. Variable coefficients of the equation can affect the solitonic structure, when they are specially chosen, while curved and linear solitons are illustrated. Elastic collisions between/among two and three solitons are discussed, through which the solitons keep their original shapes invariant except for some phase shifts.     

Elliptic incoherent accessible solitons in strongly nonlocal media

We study the propagation of elliptic incoherent accessible solitons in strongly nonlocal media with noninstantaneous Kerr nonlinearity. For this soliton to exist, the coherence properties of the incoherent beam should be anisotropic. The total power of the incident beam should also equal to a critical value which depend on the beam width as well as the coherence properties. When initial parameters of the beam do not satisfy the existence conditions, the elliptic incoherent accessible solitons will undergo linear harmonic oscillation in different states. Corresponding properties are studied in detail.     

Waveguides induced by steady-state gray solitons in biased photorefractive-photovoltaic crystals

We carry out a theoretical investigation of the properties of waveguides induced by photorefractive one-dimensional steady-state gray spatial solitons (i.e., screening solitons, photovoltaic solitons, and screening-photovoltaic solitons). We demonstrate that waveguides induced by photorefractive steady-state gray spatial solitons are only a single guided mode for both all soliton graynesses and all values of ρ, where ρ is the ratio between the soliton peak intensity and the dark irradiance, and moreover, waveguides induced by gray photovoltaic solitons for closed-circuit condition are also only a single guided mode for all electric current densities. We find that the confined energy near the center of a photorefractive steady-state gray spatial soliton increases with ρ and decreases with an increase in the soliton grayness. We also find that the confined energy near the center of a gray photovoltaic soliton for closed-circuit condition increases with the electric current density. On the other hand, waveguides induced by gray screening-photovoltaic solitons are gray screening soliton-induced waveguides when the bulk photovoltaic effect is neglectable and are gray photovoltaic soliton-induced waveguides when the external bias field is absent.     

椭圆响应强非局域非线性介质中的二维异步分数傅里叶变换及光束传输特性

研究了椭圆响应强非局域非线性介质中的光束传输问题. 结果表明:任意光束在这类介质中传输时均遵守二维异步分数傅里叶变换的传输规律. 基于二维异步分数傅里叶变换这一数学工具, 可很方便地对光束的传输进行解析求解并分析其性质. 利用二维异步分数傅里叶变换的性质, 讨论了一般光束的传输性质; 分析了孤子和二维异步呼吸子的形成条件; 得出了孤子/呼吸子的相互作用规律. 关键词： 椭圆响应 强非局域非线性 孤子 呼吸子     

Regimes of operation states in passively mode-locked fiber soliton ring laser

The principal of passively mode-locked fiber soliton ring lasers is summarized, including its three output operation states: normal soliton, bound–solitons and noise-like pulse. The experimental results of the passively mode-locked fiber soliton ring lasers developed by us are given. Bound–solitons with different discrete separations and three-bound–solitons state have been observed in our fiber laser for the first time. The relationship among three operation states in fiber soliton laser is analyzed.     

Bäcklund transformation,infinitely-many conservation laws,solitary and periodic waves of an extended (3 + 1)-dimensional Jimbo–Miwa equation with time-dependent coefficients

In this paper, an extended (3+1)-dimensional Jimbo–Miwa equation with time-dependent coefficients is investigated, which comes from the second member of the Kadomtsev–Petviashvili hierarchy and is shown to be conditionally integrable. Bilinear form, Bäcklund transformation, Lax pair and infinitely-many conservation laws are derived via the binary Bell polynomials and symbolic computation. With the help of the bilinear form, one-, two- and three-soliton solutions are obtained via the Hirota method, one-periodic wave solutions are constructed via the Riemann theta function. Additionally, propagation and interaction of the solitons are investigated analytically and graphically, from which we find that the interaction between the solitons is elastic and the time-dependent coefficients can affect the soliton velocities, but the soliton amplitudes remain unchanged. One-periodic waves approach the one-solitary waves with the amplitudes vanishing and can be viewed as a superposition of the overlapping solitary waves, placed one period apart.