Fractal dependence of vector-soliton collisions in birefringent fibers

Vector-soliton collisions in birefringent nonlinear optical fibers are investigated. The underlying mathematical model is the non-integrable coupled nonlinear Schrödinger equations. It is shown that the exit velocity versus collision velocity graph has a fractal structure. When we zoom into different positions of this fractal, we get structures which are either a copy, a horizontal reflection or vertical reflection of the original structure. Collision dynamics in the zoomed-in windows and that in the original graph follow simple and well-defined patterns as well. We explain this fractal dependence of the collision by a novel resonance mechanism between translational motion of vector solitons and radiation modes which cause internal oscillations inside a vector soliton.     

孤子的碰撞——带外场的非线性Shrödinger方程的孤子之间的碰撞

本文用计算机考察了带外场(即变系数)非线性Shrödinger方程的孤子间碰撞机制。发现这种作变速运动的孤子碰后仍能恢复各自的形状。近年来,对孤立子的研究已深入到带外场的非线性Shrödinger方程。     

PARTICULAR SOLITONS IN NONLINEAR LATTICE

One soliton of particle velocity and its amplitude (maximum particle velocity of one soliton) in Toda lattice is given analytically. It has also been known numerically that the maximum particle velocity (when the collision of two solitons reaches their maximum, we define V_n at this time as its maximum particle velocity) during the collision of two solitons moving in the same direction is equal to the difference between the amplitudes of two solitons if the difference is large enough; however, the maximum particle velocity is equal to the adding up of the amplitudes of two solitons moving in the opposite directions. The relationship between the maximum value of e^-(r_n)-1 and their initial amplitude of e^-(r_n)-1 is also given analytically in Toda lattice if the amplitudes of the two solitons are the same and their moving directions are opposite. Compared with the Boussinesq equation, there are differences between the Toda lattice equation and the Boussinesq equation for solitons during the collision.     

Interaction of oblique dark solitons in two-dimensional supersonic nonlinear Schrödinger flow

We investigate the collision of two oblique dark solitons in the two-dimensional supersonic nonlinear Schrödinger flow past two impenetrable obstacles. We numerically show that this collision is very similar to the dark solitons collision in the one-dimensional case. We observe that it is practically elastic and we measure the shifts of the solitons positions after their interaction.     

(2+1)维非对称 Nizhnik-Novikov-Veselov系统的新映射解及其局域结构

映射法是一种非常经典、有效和成熟的求解非线性演化方程的方法，其最大的特点是可以有多种不同形式的设解，使得最终求得的解丰富多彩. 利用改进的 Riccati 方程映射法和变量分离法，得到了(2+1)维非对称 Nizhnik-Novikov-Veselov 系统的新显式精确解.根据得到的孤波解，构造出该系统的峰孤子和分形孤子等局域结构，研究了两个孤立波的“追碰”现象. 关键词： 改进的映射法 (2+1)维非对称 Nizhnik-Novikov-Veselov 系统 局域结构 “追碰”现象     

Collision characteristics between two temporal untraslow vector optical solitons in a five-level V type system

By using a multiple-scale method, we study the formation of the temporal ultraslow vector optical solitons (USVOS) in a cold lifetime-broadened five-level V type atomic system via electromagnetically induced transparency. And we find that the two orthogonally polarized components of the weak pulsed probe field can evolve into a pair of temporal USVOS. Subsequently, by numerical simulating the collision between two temporal USVOS, we obtain that their collision properties are correlated with their incident angle and initial phase shift. Whether two solitons are in phase or out of phase, their collisions are almost elastic. Especially, when two solitons are parallel input, the collisions between them are periodical. Our results may have potential application in optical information processing and engineering.     

Collision between scalar and vector spatial solitons in Kerr media

Experimental observation and numerical results concerning collisions between scalar and vector spatial solitons in a Kerr planar waveguide are presented. It is shown that this configuration allows for the full control of spatial and polarization dynamics of the interacting vector solitons. On the one hand, the ability to achieve polarization control of a single-hump vector soliton is demonstrated. On the other hand, the effect of collision on the spatial symmetry-breaking dynamics of multimode vector solitons is investigated.     

爱因斯坦方程的一组严格的双孤立波解

由一个对角型的度规出发,给出了一组严格的引力双孤立波解。这组解是处处正则和稳定的,它们描写两个分别沿正z方向和负z方向的光速传播的孤立波,而且在彼此碰撞后不会改变自己形状并继续传播,文章最后还讨论了这组解相应的时空奇异性和能量正定性问题。 关键词：     

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.     

Interference properties of two-component matter wave solitons

Wave properties of solitons in a two-component Bose-Einstein condensate are investigated in detail.We demonstrate that dark solitons in one of components admit interference and tunneling behavior,in sharp contrast to the scalar dark solitons and vector dark solitons.Analytic analyses of interference properties show that spatial interference patterns are determined by the relative velocity of solitons,while temporal interference patterns depend on the velocities and widths of two solitons,differing from the interference properties of scalar bright solitons.Especially,for an attractive interactions system,we show that interference effects between the two dark solitons can induce some short-time density humps(whose densities are higher than background density).Moreover,the maximum hump value is remarkably sensitive to the variation of the solitons' parameters.For a repulsive interactions system,the temporal-spatial interference periods of dark-bright solitons have lower limits.Numerical simulation results suggest that interference patterns for the dark-bright solitons are more robust against noises than bright-dark solitons.These explicit interference properties can be used to measure the velocities and widths of solitons.It is expected that these interference behaviors can be observed experimentally and can be used to design matter wave soliton interferometer in vector systems.     

Soliton interaction in the coupled mixed derivative nonlinear Schrödinger equations

The bright one- and two-soliton solutions of the coupled mixed derivative nonlinear Schrödinger equations in birefringent optical fibers are obtained by using the Hirota's bilinear method. The investigation on the collision dynamics of the bright vector solitons shows that there exists complete or partial energy switching in this coupled model. Such parametric energy exchanges can be effectively controlled and quantificationally measured by analyzing the collision dynamics of the bright vector solitons. The influence of two types of nonlinear coefficient parameters on the energy of each vector soliton, is also discussed. Based on the significant energy transfer between the two components of each vector soliton, it is feasible to exploit the future applications in the design of logical gates, fiber directional couplers and quantum information processors.     

Transition,coexistence, and interaction of vector localized waves arising from higher-order effects

We study vector localized waves on continuous wave background with higher-order effects in a two-mode optical fiber. The striking properties of transition, coexistence, and interaction of these localized waves arising from higher-order effects are revealed in combination with corresponding modulation instability (MI) characteristics. It shows that these vector localized wave properties have no analogues in the case without higher-order effects. Specifically, compared to the scalar case, an intriguing transition between bright–dark rogue waves and w-shaped–anti-w-shaped solitons, which occurs as a result of the attenuation of MI growth rate to vanishing in the zero-frequency perturbation region, is exhibited with the relative background frequency. In particular, our results show that the w-shaped–anti-w-shaped solitons can coexist with breathers, coinciding with the MI analysis where the coexistence condition is a mixture of a modulation stability and MI region. It is interesting that their interaction is inelastic and describes a fusion process. In addition, we demonstrate an annihilation phenomenon for the interaction of two w-shaped solitons which is identified essentially as an inelastic collision in this system.     

Tunneling-induced ultraslow bright and dark solitons in quantum wells

We show the formation of tunneling-induced ultraslow bright and dark solitons in an asymmetric double-quantumwell structure based on the tunneling induced transparency.In this semiconductor structure,the pump field is replaced by the electron-tunneling coupling,which can be modulated by a static electric field.With appropriate conditions,we demonstrate by modulating the intensity of the static electric field that the interplay between the group velocity dispersion and the self-Kerr nonlinearity results in the generation of dark and bright solitons with ultraslow group velocity.     

Splitting of strain solitons upon their interaction

The results from analytical studies and numerical simulations of strain soliton interaction are presented. It is shown that soliton collisions are of an inelastic character; i.e., secondary (daughter) solitons with lower energies are produced after a collision, and either the residual energy is spent in a wave process, or additional secondary solitons can form if the energy is sufficient. Qualitatively different scenarios of this interaction depend on the relative velocity of collision. Elements of visualizing nonlinear wave processes via shadowgraphy are discussed.     

Observation of temporal vector soliton propagation and collision in birefringent fiber

We report the experimental observation of temporal vector soliton propagation and collision in a linearly birefringent optical fiber. To the best of the authors' knowledge, this is both the first demonstration of temporal vector solitons with two mutually incoherent component fields, and of vector soliton collisions in a Kerr nonlinear medium. Collisions are characterized by an intensity redistribution between the two components, and the experimental results agree with numerical predictions of the coupled nonlinear Schr?dinger equation.     

Dark and anti-dark vector solitons of the coupled modified nonlinear Schrödinger equations from the birefringent optical fibers

Coupled modified nonlinear Schr?dinger (CMNLS) equations describe the pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers. A new type of the Lax pair and another hierarchy of the infinitely many conservation laws are derived based on the Wadati-Konno-Ichikawa system. By means of the Hirota method, soliton solutions in the normal dispersion regime are obtained. Parametric regions for the existence of dark and anti-dark vector soliton solutions are given. Asymptotic analysis shows that the collision between two solitons (two anti-dark solitons, two dark solitons, or dark and anti-dark solitons) in each polarization direction is elastic. Moreover, there is no energy transfer between two polarization components of each vector soliton, whether dark or anti-dark vector soliton. In addition, dark and anti-dark solitons can coexist on the same background seen from the collision between the dark and anti-dark solitons in one polarization direction. Our graphical analysis shows that the parameters in the CMNLS equations not only determine the regions for the existence of dark and anti-dark soliton solutions but also control the phase and direction of the propagation of the solitons. Finally, through the linear stability analysis, the modulational instability condition is given.     

Superposition solitons in two-component Bose-Einstein condensates

We develop the Hirota bilinear method and obtain the exact one and two superposition soliton solutions for two-component Bose-Einstein condensates. The conversion of three kinds of solitons including the superposition solitons, bright-bright solitons, and dark-bright solitons is discussed. With the energy analysis, we find that the superposition soliton state is an excitation state for this system. Moreover, the collision of two superposition solitons is found to be elastic.     

Multipole spatial vector solitons

We introduce the concept of multipole spatial optical vector solitons associated with higher-order guided modes trapped by a soliton-induced waveguide in a bulk medium. Such stationary localized waves include previously predicted vortex- and dipole-mode vector solitons and also describe new higher-order vector solitons and necklace-type beams. We present the theoretical and experimental results of the structure, formation, and instability development of the quadrupole vector solitons.     

Structure and evolution of strange attractors in non-elastic triangular billiards

We study non-elastic billiard dynamics in an equilateral triangular table. In such dynamics, collisions with the walls of the table are not elastic, as in standard billiards; rather, the outgoing angle of the trajectory with the normal vector to the boundary at the point of collision is a uniform factor λ < 1 smaller than the incoming angle. This leads to contraction in phase space for the discrete-time dynamics between consecutive collisions, and hence to attractors of zero Lebesgue measure, which are almost always fractal strange attractors with chaotic dynamics, due to the presence of an expansion mechanism. We study the structure of these strange attractors and their evolution as the contraction parameter λ is varied. For λ∈(0,1/3), we prove rigorously that the attractor has the structure of a Cantor set times an interval, whereas for larger values of λ gaps arise in the Cantor structure. For λ close to 1, the attractor splits into three transitive components, whose basins of attraction have fractal boundaries.     

Nonlinear photonic crystals near the supercollimation point

We uncover a strong coupling between nonlinearity and diffraction in a photonic crystal at the supercollimation point. We show that this is modeled by a nonlinear diffraction term in a nonlinear-Schr?dinger-type equation in which the properties of solitons are investigated. Linear stability analysis shows solitons are stable in an existence domain that obeys the Vakhitov-Kolokolov criterium. In addition, we investigate the influence of the nonlinear diffraction on soliton collision scenarios.