Collisions between discrete spatiotemporal dissipative Ginzburg-Landau solitons in two-dimensional photonic lattices

Systematic results of collisions between discrete spatiotemporal dissipative Ginzburg-Landau solitons in two-dimensional photonic lattices are reported. The generic outcomes are identified for (i) the collision of two identical solitons located in the corner, at the edge, and in the center of the photonic lattice, and for (ii) the collision of two non-identical corner and edge solitons located at different distances from the boundaries of the photonic lattice. Depending on the values of the kick (collision momentum) and of the nonlinear (cubic) gain, the collision scenarios include soliton merging, creation of an extra soliton, soliton bouncing, soliton spreading, and quasi-elastic (symmetric) interactions.     

Discrete light bullets in two-dimensional photonic lattices: Collision scenarios

We report systematic results of collisions between discrete spatiotemporal optical solitons in two-dimensional photonic lattices. We show that the outcomes of collisions strongly depend on the initial soliton parameters, such as their input amplitudes (energies) and their transverse velocities. Four generic outcomes are identified in the study of collisions between discrete light bullets located in the corner, at the edge, and in the center of the photonic lattice: (a) merger of both low and high amplitude solitons into a single one, at small values of the kick parameter (soliton transverse velocity), (b) spreading of low amplitude solitons at intermediate values of the kick parameter, (c) bouncing of high amplitude solitons at intermediate values of the kick parameter, which is accompanied by a sharp modification of input soliton transverse velocities, and (d) quasi-elastic (symmetric) interactions of both low and high amplitude solitons at large values of the kick parameter.     

Collisions between spinning and nonspinning co-axial three-dimensional Ginzburg-Landau solitons

We report results of the first analysis of collisions between stable fundamental (alias spinless) and vortical (spinning) three-dimensional dissipative solitons in a model of a laser cavity. The systematic analysis is carried out for values S=1 and S=2 of the vorticity of the latter soliton. With the increase of the collision momentum, Χ, the same generic scenarios are observed in either case: merger into a single fundamental soliton at both small and relatively large values of Χ, and the formation of two fundamental solitons in an intermediate interval of variation of the collision momentum Χ. At very large values of Χ, the collision seems quasi-elastic, but the vortex soliton eventually splits into two nonspinning fragments.     

Collisions between type II two-dimensional quadratic solitons

We report experimental and numerical results that describe collisions between two-dimensional type II quadratic solitons excited in a KTP crystal by fundamental waves of orthogonal polarization. Our results provide experimental evidence of the possibility of both inelastic collision (when two quadratic solitons merge at input into a single soliton at output) and quasi-elastic collision.     

Interaction of two-space-dimensional classical Q solitons

The stability region and the head-on collisions of two-space-dimensional “charged” (x, y, t) solitons have been investigated via a computer in the framework of the Lorentz-invariant Klein-Gordon equation with saturable nonlinearity. The range of parameters has been determined where (i) quasi-elastic soliton interaction occurs and (ii) the formation of two soliton bound states takes place. The collision of solitons may lead to their decay.     

Families of Bragg-grating solitons in a cubic–quintic medium

We investigate the existence and stability of solitons in an optical waveguide equipped with a Bragg grating (BG) in which nonlinearity contains both cubic and quintic terms. The model has straightforward realizations in both temporal and spatial domains, the latter being most realistic. Two different families of zero-velocity solitons, which are separated by a border at which solitons do not exist, are found in an exact analytical form. One family may be regarded as a generalization of the usual BG solitons supported by the cubic nonlinearity, while the other family, dominated by the quintic nonlinearity, includes novel “two-tier” solitons with a sharp (but nonsingular) peak. These soliton families also differ in the parities of their real and imaginary parts. A stability region is identified within each family by means of direct numerical simulations. The addition of the quintic term to the model makes the solitons very robust: simulating evolution of a strongly deformed pulse, we find that a larger part of its energy is retained in the process of its evolution into a soliton shape, only a small share of the energy being lost into radiation, which is opposite to what occurs in the usual BG model with cubic nonlinearity.     

Collision Dynamics of Dissipative Matter-Wave Solitons in a Perturbed Optical Lattice

We investigate the stability and collision dynamics of dissipative matter-wave solitons formed in a quasi-onedimensional Bose-Einstein condensate with linear gain and three-body recombination loss perturbed by a weak optical lattice.It is shown that the linear gain can modify the stability of the single dissipative soliton moving in the optical lattice.The collision dynamics of two individual dissipative matter-wave solitons explicitly depend on the linear gain parameter,and they display different dynamical behaviors in both the in-phase and out-of-phase interaction regimes.     

Mobility of discrete solitons in quadratically nonlinear media

We study the mobility of solitons in lattices with quadratic (chi(2), alias second-harmonic-generating) nonlinearity. Using the notion of the Peierls-Nabarro potential and systematic numerical simulations, we demonstrate that, in contrast with their cubic (chi(3)) counterparts, the discrete quadratic solitons are mobile not only in the one-dimensional (1D) setting, but also in two dimensions (2D), in any direction. We identify parametric regions where an initial kick applied to a soliton leads to three possible outcomes: staying put, persistent motion, or destruction. On the 2D lattice, the solitons survive the largest kick and attain the largest speed along the diagonal direction.     

脉冲内拉曼散射影响下的孤子之间的相互作用及其控制

研究了脉冲内拉曼散射效应影响下的同相和反相相邻孤子脉冲之间的相互作用,分析了孤子之间的相互作用对定时抖动的影响和脉冲内拉曼散射效应对孤子频移的影响。研究结果表明:在脉冲内,在拉曼散射效应的影响下,同相基态孤子脉冲的周期性离合被破坏了,两孤子脉冲一次碰撞后一直处于排斥状态,并且在碰撞后自频移现象十分明显;反相孤子脉冲的影响则较弱,两孤子脉冲都向下降沿发生偏移。引入非线性增益可以有效地控制孤子之间的相互作用,抑制自频移效应和稳定孤子传输。     

Numerical analysis of the stability of optical bullets (2 + 1) in a planar waveguide with cubic–quintic nonlinearity

In this paper, we have presented a numerical analysis of the stability of optical bullets (2 + 1), or spatiotemporal solitons (2 + 1), in a planar waveguide with cubic–quintic nonlinearity. The optical spatiotemporal solitons are the result of the balance between the nonlinear parameters, of dispersion (dispersion length, L _D) and diffraction (diffraction length, L _d) with temporal and spatial auto-focusing behavior, respectively. With the objective of ensure the stability and preventing the collapse or the spreading of pulses, in this study we explore the cubic–quintic nonlinearity with the optical fields coupled by cross-phase modulation and considering several values for the non linear parameter α We have shown the existence of stable light bullets in planar waveguide with cubic–quintic nonlinearity through the study of spatiotemporal collisions of the light bullets.     

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.     

Multisoliton solutions and energy sharing collisions in coupled nonlinear Schrödinger equations with focusing,defocusing and mixed type nonlinearities

Bright and bright-dark type multisoliton solutions of the integrable N-coupled nonlinear Schrödinger (CNLS) equations with focusing, defocusing and mixed type nonlinearities are obtained by using Hirota’s bilinearization method. Particularly, for the bright soliton case, we present the Gram type determinant form of the n-soliton solution (n:arbitrary) for both focusing and mixed type nonlinearities and explicitly prove that the determinant form indeed satisfies the corresponding bilinear equations. Based on this, we also write down the multisoliton form for the mixed (bright-dark) type solitons. For the focusing and mixed type nonlinearities with vanishing boundary conditions the pure bright solitons exhibit different kinds of nontrivial shape changing/energy sharing collisions characterized by intensity redistribution, amplitude dependent phase-shift and change in relative separation distances. Due to nonvanishing boundary conditions the mixed N-CNLS system can admit coupled bright-dark solitons. Here we show that the bright solitons exhibit nontrivial energy sharing collision only if they are spread up in two or more components, while the dark solitons appearing in the remaining components undergo mere standard elastic collisions. Energy sharing collisions lead to exciting applications such as collision based optical computing and soliton amplification. Finally, we briefly discuss the energy sharing collision properties of the solitons of the (2+1) dimensional long wave-short wave resonance interaction (LSRI) system.     

Interactions of solitons with complex defects in Bragg gratings

We examine collisions of moving solitons in a fiber Bragg grating with a triplet composed of two closely set repulsive defects of the grating and an attractive one inserted between them. A doublet (dipole), consisting of attractive and repulsive defects with a small distance between them, is considered too. Systematic simulations demonstrate that the triplet provides for superior results, as concerns the capture of a free pulse and creation of a standing optical soliton, in comparison with recently studied traps formed by single and paired defects, as well as the doublet: 2/3 of the energy of the incident soliton can be captured when its velocity attains half the light speed in the fiber (the case most relevant to the experiment), and the captured soliton quickly relaxes to a stationary state. A subsequent collision between another free soliton and the pinned one is examined too, demonstrating that the impinging soliton always bounces back, while the pinned one either remains in the same state, or is kicked out forward, depending on the collision velocity and phase shift between the solitons.     

Interaction of spatial photorefractive solitons in a planar waveguide

We report the observation of collisions between one-dimensional bright photorefractive screening solitons in a planar strontium–barium niobate waveguide. Depending on the intersection angle of the two solitons and their relative phase, we observe soliton fusion, repelling, energy exchange, and the creation of a third soliton upon interaction. Received: 4 November 1998 / Revised version: 3 December 1998 / Published online: 24 February 1999     

Self–propelled soliton collisions in a semiconductor cavity soliton laser

Spontaneous soliton motion has been demonstrated in different systems supporting cavity solitons. Here we consider the case of a semiconductor laser with an intracavity saturable absorber, and study the interactions between self-propelled solitons when two of them collide or when they hit a localised defect in the material gain. According to the soliton velocity and impact parameter, destructive or repulsive collisions may take place between travelling solitons. On the other hand, a very rich variety of dynamical behaviors can be observed when a travelling soliton hits a material defect of comparable size. We observe soliton destruction, repulsive or attractive interaction and two trapped cases. The behavior is mainly determined by the gain contrast between the defect and the background.     

二阶孤子的传输和相互作用

用分步傅里叶变换法求解二阶孤子传输的非线性薛定谔方程, 得到了在此条件下孤子传输的数值图形, 发现二阶孤子在传输中被压缩, 幅值振荡变化。2个二阶孤子在传输过程中没有出现象2个一阶孤子那样周期性碰撞, 但2个二阶孤子时间间隔较小时, 随传输距离在2个二阶孤子中间周期性地衍生出第3个孤子。研究证明：二阶孤子的传输具有与一阶孤子明显不同的特征。     

Angular surface solitons in sectorial hexagonal arrays

We report on the experimental observation of corner surface solitons localized at the edges joining planar interfaces of hexagonal waveguide array with uniform nonlinear medium. The face angle between these interfaces has a strong impact on the threshold of soliton excitation as well as on the light energy drift and diffraction spreading.     

Soliton molecules and asymmetric solitons of the extended Lax equation via velocity resonance

We investigate the techniques for velocity resonance and apply them to construct soliton molecules using two solitons of the extended Lax equation.What is more,each soliton molecule can be transformed into an asymmetric soliton by changing the parameterφ.In addition,the collision between soliton molecules(or asymmetric soliton)and several soliton solutions is observed.Finally,some related pictures are presented.     

Solitons in Bose–Einstein condensates

The Gross–Pitaevskii equation (GPE) describing the evolution of the Bose–Einstein condensate (BEC) order parameter for weakly interacting bosons supports dark solitons for repulsive interactions and bright solitons for attractive interactions. After a brief introduction to BEC and a general review of GPE solitons, we present our results on solitons that arise in the BEC of hard-core bosons, which is a system with strongly repulsive interactions. For a given background density, this system is found to support both a dark soliton and an antidark soliton (i.e., a bright soliton on a pedestal) for the density profile. When the background has more (less) holes than particles, the dark (antidark) soliton solution dies down as its velocity approaches the sound velocity of the system, while the antidark (dark) soliton persists all the way up to the sound velocity. This persistence is in contrast to the behaviour of the GPE dark soliton, which dies down at the Bogoliubov sound velocity. The energy–momentum dispersion relation for the solitons is shown to be similar to the exact quantum low-lying excitation spectrum found by Lieb for bosons with a delta-function interaction.     

Stabilization of the Soliton Transported Bio-energy in Protein Molecules in the Improved Model

We study the stabilization of the soliton transported bio-energy by the dynamic equations in the improved Davydov theory from four aspects containing the feature of free motion and states of the soliton at the long-time motion and at biological temperature 300 K and behaviors of collision of the solitons by Runge-Kutta method and physical parameter values appropriate to the $\alpha$-helix protein molecules. We prove that the new solitons can move without dispersion at a constant speed retaining its shape and energy in free and long-time motions and can go through each other without scattering. If considering further influence of the temperature effect of heat bath on the soliton, it is still thermally stable at biological temperature 300 K and in a time as long as 300 ps and amino acid spacings as large as 400, which shows that the lifetime of the new soliton is at least 300 ps, which is consistent with analytic result obtained by quantum perturbation theory. These results exhibit that the new soliton is a possible carrier of bio-energy transport and the improved model is possibly a candidate for the mechanism of this transport.