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.     

Bogomol'nyi solitons and Hermitian symmetric spaces

We apply the coadjoint orbit method to construct relativistic nonlinear sigma models (NLSM) on the target space of coadjoint orbits coupled with the Chern-Simons (CS) gauge field and we study self-dual solitons. When the target space is given by a Hermitian symmetric space (HSS), we find that the system admits self-dual solitons whose energy is Bogomol'nyi bounded from below by a topological charge. The Bogomol'nyi potential on the Hermitian symmetric space is obtained in the case when the maximal torus subgroup is gauged, and the self-dual equation in the CP(N − 1) case is explored. We also discuss the self-dual solitons in the case of noncompact SU(1, 1) and present a detailed analysis for the rotationally symmetric solutions.     

Weierstrass \wp -lumps

We present our results of a numerical investigation of the behaviour of a system of two solitons in the (2+1) dimensional CP¹ model on a torus. Defined by the elliptic function of Weierstrass, and working in the Skyrme version of the model, the soliton lumps exhibit splitting, scattering at right angles and motion reversal in the various configurations considered. The work is restricted to systems with no initial velocity. Received 24 April 2001     

Torus destruction via global bifurcations in a piecewise-smooth, continuous map with square-root nonlinearity

It has been shown recently that torus formation in piecewise-smooth maps can occur through a special type of border collision bifurcation in which a pair of complex conjugate Floquet multipliers “jump” from the inside to the outside of the unit circle. It has also been shown that a large class of impacting mechanical systems yield piecewise-smooth maps with square-root singularity. In this Letter we investigate the dynamics of a two-dimensional piecewise-smooth map with square-root type nonlinearity, and describe two new routes to chaos through the destruction of two-frequency torus. In the first scenario, we identify the transition to chaos through the destruction of a loop torus via homoclinic bifurcation. In the other scenario, a change of structure in the torus occurs via heteroclinic saddle connections. Further parameter changes lead to a homoclinic bifurcation resulting in the creation of a chaotic attractor. However, this scenario is much more complex, with the appearance of a sequence of heteroclinic and homoclinic bifurcations.     

Topological quantum liquids with quaternion non-Abelian statistics

Noncollinear magnetic order is typically characterized by a tetrad ground state manifold (GSM) of three perpendicular vectors or nematic directors. We study three types of tetrad orders in two spatial dimensions, whose GSMs are SO(3) = S(3)/Z(2), S(3)/Z(4), and S(3)/Q(8), respectively. Q(8) denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disordering these three types of tetrad orders, the systems enter fully gapped liquid phases described by Z(2), Z(4), and non-Abelian quaternion gauge field theories, respectively. The latter case realizes Kitaev's non-Abelian toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This non-Abelian topological phase possesses a 22-fold ground state degeneracy on the torus arising from the 22 representations of the Drinfeld double of Q(8).     

Matter-wave bright solitons in effective bichromatic lattice potentials

Matter-wave bright solitons in bichromatic lattice potentials are considered and their dynamics for different lattice environments are studied. Bichromatic potentials are created from superpositions of (i) two linear optical lattices and (ii) a linear and a nonlinear optical lattice. Effective potentials are found for the solitons in both bichromatic lattices and a comparative study is done on the dynamics of solitons with respect to the effective potentials. The effects of dispersion on solitons in bichromatic lattices are studied and it is found that the dispersive spreading can be minimized by appropriate combinations of lattice and interaction parameters. Stability of nondispersive matter-wave solitons is checked from phase portrait analysis.     

二维线性与非线性光晶格中物质波孤立子的稳定性

利用变分法和数值计算方法研究了二维线性和非线性光晶格中二维玻色-爱因斯坦凝聚体系中物质波孤立子的存在及其稳定性. 利用定态变分原理及Vakhitov-Kolokolov判据总结了线性和非线性结合光晶格中几种参数组合下定态定域解的稳定性. 结果表明, 当存在二维非线性光晶格时, 在吸引和排斥相互作用的原子体系中均可以存在稳定的物质波孤立子. 另外, 利用含时变分法研究了线性和非线性光晶格中物质波孤立子随时间的传播特性, 使波包参数对时间的一阶导数等于零, 可以给出稳定状态对应的参数, 结论和定态变分法给出的结果一致. 最后用数值计算方法研究变分结果的正确性, 把变分结果作为初始条件代入Gross-Pitaevskii方程研究其随时间传播特征, 得到了稳定的传播过程, 所得到的结果和变分分析结果一致. 关键词： 线性非线性光晶格 玻色-爱因斯坦凝聚 孤立子 稳定性     

Composite multihump vector solitons carrying topological charge

We propose composite solitons carrying topological charge: multicomponent two dimensional [ (2+1)D] vector (Manakov-like) solitons for which at least one component carries topological charge. These multimode solitons can have a single hump or exhibit a multihump structure. The "spin" carried by these multimode composite solitons suggests 3D soliton interactions in which the particlelike behavior includes spin, in addition to effective mass, linear, and angular momenta.     

强非局域介质中多个空间孤子的相互作用

研究了在强非局域情况下,非局域非线性介质中多个(大于两个)空间孤子相互作用的特点与规律。以Snyder-Mitchell线性模型为理论基础并利用线性叠加原理来构建解。另外,还采用了将孤子作为粒子处理的方法。主要考虑空间斜对称入射的三、四束光的相互作用,从而得出多个空间孤子相互作用的规律。还得出了(1 2)维多孤子相互作用的精确解析解,并且利用解析解画出了多孤子传输过程中的光强分布图。根据解析解发现,多孤子能形成稳定的束缚态向前传输,缠绕与否与初始入射方向有关;作用过程中并无能量转移,且相互作用与初始相位无关。将孤子作为粒子处理得到的孤子相互作用的规律与解析解得出的规律一致。     

Birth of bilayered torus and torus breakdown in a piecewise-smooth dynamical system

Border-collision bifurcations arise when the periodic trajectory of a piecewise-smooth system under variation of a parameter crosses into a region with different dynamics. Considering a three-dimensional map describing the behavior of a DC/DC power converter, the Letter discusses a new type of border-collision bifurcation that leads to the birth of a “bilayered torus”. This torus consists of the union of two saddle cycles, their unstable manifolds, and a stable focus cycle. When changing the parameters, the bilayered torus transforms through a border-collision bifurcation into a resonance torus containing the stable cycle and a saddle. The Letter also presents scenarios for torus destruction through homoclinic and heteroclinic tangencies.     

Discrete solitons in zigzag waveguide arrays with different types of linear mixing between nearest-neighbor and next-nearest-neighbor couplings

We study discrete solitons in zigzag discrete waveguide arrays with different types of linear mixing between nearest-neighbor and next-nearest-neighbor couplings. The waveguide array is constructed from two layers of one-dimensional (1D) waveguide arrays arranged in zigzag form. If we alternately label the number of waveguides between the two layers, the cross-layer couplings (which couple one waveguide in one layer with two adjacent waveguides in the other layer) construct the nearest-neighbor couplings, while the couplings that couple this waveguide with the two nearest-neighbor waveguides in the same layer, i.e., self-layer couplings, contribute the next-nearest-neighbor couplings. Two families of discrete solitons are found when these couplings feature different types of linear mixing. As the total power is increased, a phase transition of the second kind occurs for discrete solitons in one type of setting, which is formed when the nearest-neighbor coupling and next-nearest-neighbor coupling feature positive and negative linear mixing, respectively. The mobilities and collisions of these two families of solitons are discussed systematically throughout the paper, revealing that the width of the soliton plays an important role in its motion. Moreover, the phase transition strongly influences the motions and collisions of the solitons.     

Two dimensional counterpropagating spatial solitons in photorefractive crystals

We demonstrate experimentally the existence of two transverse-dimensional counterpropagating (CP) incoherent spatial solitons in a 5 x 5 x 23 mm SBN:60Ce photorefractive crystal and investigate their dynamical behavior. We carry out numerical simulations that confirm our experimental findings. Substantially different behavior from the copropagating incoherent solitons is found. A symmetry breaking transition from stable overlapping CP solitons to unstable transversely displaced CP solitons is observed. We perform linear stability analysis that predicts the threshold for the split-up transition, in qualitative agreement with numerical simulations and experimental results.     

Soliton dynamics in the complex modified KdV equation with a periodic potential

We study the existence and stability of stationary and moving solitary waves in a periodically modulated system governed by an extended cmKdV (complex modified Korteweg-de Vries) equation. The proposed equation describes, in particular, the co-propagation of two electromagnetic waves with different amplitudes and orthogonal linear polarizations in a liquid crystal waveguide, the stronger (nonlinear) wave actually carrying the soliton, while the other (a nearly linear one) creates an effective periodic potential. A variational analysis predicts solitons pinned at minima and maxima of the periodic potential, and the Vakhitov-Kolokolov criterion predicts that some of them may be stable. Numerical simulations confirm the existence of stable stationary solitary waves trapped at the minima of the potential, and show that persistently moving solitons exist too. The dynamics of pairs of interacting solitons is also studied. In the absence of the potential, the interaction is drastically different from the behavior known in the NLS (nonlinear Schrödinger) equation, as the force of the interaction between the cmKdV solitons is proportional to the sine, rather than cosine, of the phase difference between the solitons. In the presence of the potential, two solitons placed in one potential well form a persistently oscillating bound state.     

Multilayered tori in a system of two coupled logistic maps

The Letter describes different mechanisms for the formation and destruction of tori that are formed as layered structures of several sets of interlacing manifolds, each with their associated stable and unstable resonance modes. We first illustrate how a three layered torus can arise in a system of two coupled logistic maps through period-doubling or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We hereafter present two different scenarios by which a multilayered torus can be destructed. One scenario involves a cascade of period-doubling bifurcations of both the stable and the saddle cycles, and the second scenario describes a transition in which homoclinic bifurcations destroy first the two outer layers and thereafter also the inner layer of a three-layered torus. It is suggested that the formation of multilayered tori is a generic phenomenon in non-invertible maps.     

Discrete gap solitons in a diffraction-managed waveguide array

A model including two nonlinear chains with linear and nonlinear couplings between them, and opposite signs of the discrete diffraction inside the chains, is introduced. In the case of the cubic [ χ⁽³⁾] nonlinearity, the model finds two different interpretations in terms of optical waveguide arrays, based on the diffraction-management concept. A continuum limit of the model is tantamount to a dual-core nonlinear optical fiber with opposite signs of dispersions in the two cores. Simultaneously, the system is equivalent to a formal discretization of the standard model of nonlinear optical fibers equipped with the Bragg grating. A straightforward discrete second-harmonic-generation [ χ⁽²⁾] model, with opposite signs of the diffraction at the fundamental and second harmonics, is introduced too. Starting from the anti-continuum (AC) limit, soliton solutions in the χ⁽³⁾ model are found, both above the phonon band and inside the gap. Solitons above the gap may be stable as long as they exist, but in the transition to the continuum limit they inevitably disappear. On the contrary, solitons inside the gap persist all the way up to the continuum limit. In the zero-mismatch case, they lose their stability long before reaching the continuum limit, but finite mismatch can have a stabilizing effect on them. A special procedure is developed to find discrete counterparts of the Bragg-grating gap solitons. It is concluded that they exist at all the values of the coupling constant, but are stable only in the AC and continuum limits. Solitons are also found in the χ⁽²⁾ model. They start as stable solutions, but then lose their stability. Direct numerical simulations in the cases of instability reveal a variety of scenarios, including spontaneous transformation of the solitons into breather-like states, destruction of one of the components (in favor of the other), and symmetry-breaking effects. Quasi-periodic, as well as more complex, time dependences of the soliton amplitudes are also observed as a result of the instability development. Received 14 September 2002 / Received in final form 4 February 2003 Published online 24 April 2003 RID="a" ID="a"e-mail: malomed@eng.tau.ac.il     

线性塞曼劈裂对自旋-轨道耦合玻色-爱因斯坦凝聚体中亮孤子动力学的影响

利用变分近似及基于Gross-Pitaevskii方程的直接数值模拟方法,研究了自旋-轨道耦合玻色-爱因斯坦凝聚体中线性塞曼劈裂对亮孤子动力学的影响,发现线性塞曼劈裂将导致体系具有两个携带有限动量的静态孤子,以及它们在微扰下存在一个零能的Goldstone激发模和一个频率与线性塞曼劈裂有关的谐振激发模.同时给出了描述孤子运动的质心坐标表达式,发现线性塞曼劈裂明显影响孤子的运动速度和振荡周期.     

Torus destruction in a nonsmooth noninvertible map

We consider here a nonsmooth noninvertible map and report new route to chaos from a resonance loop torus which is not homeomorphic to circle but only endomorphic to it. We have found that cusp torus cannot develop before the onset of chaos, though the loop torus appears. The destruction of the loop torus occurs through homoclinic bifurcation in the presence of an infinite number of nonsmooth loops. We show that owing to the nonsmooth noninvertible nature of the map, the stable sets can bifurcate to form nonsmooth closed loops. However, that cannot be interpreted directly as basin bifurcation.     

Flights in a pseudo-chaotic system

We consider the problem of transport in a one-parameter family of piecewise rotations of the torus, for rotation number approaching 1∕4. This is a zero-entropy system which in this limit exhibits a divided phase space, with island chains immersed in a "pseudo-chaotic" region. We identify a novel mechanism for long-range transport, namely the adiabatic destruction of accelerator-mode islands. This process originates from the approximate translational invariance of the phase space and leads to long flights of linear motion, for a significant measure of initial conditions. We show that the asymptotic probability distribution of the flight lengths is determined by the geometric properties of a partition of the accelerator-mode island associated with the flight. We establish the existence of flights travelling distances of order O(1) in phase space. We provide evidence for the existence of a scattering process that connects flights travelling in opposite directions.     

Novel routes to chaos through torus breakdown in non-invertible maps

The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.