Effect of Time-Dependent Atomic Scattering Length on Solitons in Bose-Einstein Condensates with a Complex Potential

We consider the one-dimensional nonlinear Schrǒdinger equations that describe the dynamics of a Bose-Einstein Condensates with time-dependent scattering length in a complex potential. Our results show that as long as the integrable relation is satisfied, exact solutions of the one-dimensional nonlinear Schrǒdinger equation can be found in a general closed form, and interactions between two solitons are modulated in a complex potential We find that the changes of the scattering length and trapping potential can be effectively used to control the interaction between two bright soliton.     

Unified treatment of the bound states of the Schioberg and the Eckart potentials using Feynman path integral approach

We obtain analytical expressions for the energy eigenvalues of both the Schioberg and Eckart potentials using an approximation of the centrifugal term.In order to determine the e-states solutions,we use the Feynman path integral approach to quantum mechanics.We show that by performing nonlinear space-time transformations in the radial path integral,we can derive a transformation formula that relates the original path integral to the Green function of a new quantum solvable system.The explicit expression of bound state energy is obtained and the associated eigenfunctions are given in terms of hypergeometric functions.We show that the Eckart potential can be derived from the Schioberg potential.The obtained results are compared to those produced by other methods and are found to be consistent.     

Atomic population oscillations between two coupled Bose——Einstein condensates with time-dependent nonlinear interaction

The atomic population oscillations between two Bose--Einstein condensates with time-dependent nonlinear interaction in a double-well potential are studied. We first analyse the stabilities of the system's steady-state solutions. And then in the perturbative regime, the Melnikov chaotic oscillation of atomic population imbalance is investigated and the Melnikov chaotic criterion is obtained. When the system is out of the perturbative regime, numerical calculations reveal that regulating the nonlinear parameter can lead the system to step into chaos via period doubling bifurcations. It is also numerically found that adjusting the nonlinear parameter and asymmetric trap potential can result in the running-phase macroscopic quantum self-trapping (MQST). In the presence of a weak asymmetric trap potential, there exists the parametric resonance in the system.     

Phase Synchronization as a Mechanism of Controlling Spatiotemporal Chaos via External Periodic Signal

We find that phase synchronization (PS) is a mechanism in which the spatiotemporal chaos (STC) can be suppressed to a spatially regular (SR) state by applying an external periodic signal in a one-dimensional driven drift-wave system.In the driving wave coordinate,the nonlinear system can be transformed to a set of coupled oscillators moving in a periodic potential.In this multi-dimensional system,the internal modes are slaved one by one through PS by the control signal.Two types of responses of the internal modes to the external periodic signal are observed.For some modes,the stabilization is through frequency-locking;while for the other modes,a special kind of PS without frequency-locking,namely multi-looping PS,is developed.     

Transverse nonlinear vibrations of a circular spinning disk with a varying rotating speed

We analyze the transverse nonlinear vibrations of a rotating flexible disk subjected to a rotating point force with a periodically varying rotating speed. Based on Hamilton’s principle, the nonlinear governing equations of motion (coupled equations among the radial, tangential and transverse displacements) are derived for the rotating flexible disk. When the in-plane inertia is ignored and a stress function is introduced, the three nonlinearly coupled partial differential equations are reduced to two nonlinearly coupled partial differential equations. According to Galerkin’s approach, a four-degree-of-freedom nonlinear system governing the weakly split resonant modes is derived. The resonant case considered here is 1:1:2:2 internal resonance and a critical speed resonance. The primary parametric resonance for the first-order sin and cos modes and the fundamental parametric resonance for the second-order sin and cos modes are also considered. The method of multiple scales is used to obtain a set of eight-dimensional nonlinear averaged equations. Based on the averaged equations, using numerical simulations, the influence of different parameters on the nonlinear vibrations of the spinning disk is detected. It is concluded that there exist complicated nonlinear behaviors including the periodic, period-n and multi-pulse type chaotic motions for the spinning disk with a varying rotating speed. It is also found that among all parameters, the damping and excitation have great influence on the nonlinear responses of the spinning disk with a varying rotating speed.     

Analytic Study of a Bose-Einstein Condensate in Waveguide with an Obstacle Potential

We investigate the exact solutions of one-dimensional （1D） time-independent Gross-Pitaevskii equation （GPE）, which governs a Bose-Einstein condensate （BEC） in the magnetic waveguide with a square-Sech potential. Both the bound state and transmission state are found and the corresponding spatial configurations and transport properties of BEC are analyzed. It is shown that the well-known absolute transmission of the linear system can occur in the considered nonlinear system.     

Propagation properties of electromagnetic fields in elliptic dielectric hollow fibres and their applications

We numerically calculate and analyse the electromagnetic fields, optical intensity distributions, polarization states and orbital angular momentum of some elliptic hollow modes in an elliptic dielectric hollow fiber (EDHF) by using Mathieu functions, and also calculate the optical potential of the blue-detuned _eHE₁₁ mode evanescent-light wave for ⁸⁵Rb atoms, including the position-dependent van der Waals potential, and discuss briefly some potential applications of our EDHF in atom and molecule optics, etc. Our study shows that the vector electric field distributions of the odd modes in the cross section of the EDHF are the same as that of the even modes and with different boundary ellipses by rotating an angle of π/2, and the orbital angular momentum (OAM) of single HE (EH) mode is exactly equal to zero, while that of dual-mode in the EDHF is fractional in h, and has a sinusoidal oscillation as z varies. The EDHF can be used to produce various elliptic hollow beams, even to generate and study various atomic vortices with a fractional charge and its fractional quantum Hall effect in atomic Bose--Einstein condensate, and so on.     

Incoherently coupled soliton pairs in nonlocal nonlinear media

We show that incoherently coupled soliton pairs can exist in nonlocal Kerr-type nonlinear media. Such solitons can propagate in bright--bright, dark--dark, and gray--gray configurations. When the nonlocal nonlinearity is absent, these bright--bright and dark--dark soliton pairs are those observed previously in local Kerr-type nonlinear media. Our analysis indicates that for a self-focusing nonlinearity the intensity full width half maximum (FWHM) of the bright--bright pair components increases with the degree of nonlocality of the nonlinear response, whereas for a self-defocusing nonlinearity the intensity FWHM of the dark--dark and gray--gray pair components decreases with the increase in the degree of nonlocality of the nonlinear response. The stability of these soliton pairs has been investigated numerically and it has been found that they are stable.     

Longitudinal dynamics from hydrodynamics with an order parameter

We studied coupled dynamics of hydrodynamic fields and order parameter in the presence of nontrivial longitudinal flow using the chiral fluid dynamics model.We found that longitudinal expansion provides an effective relaxation for the order parameter,which equilibrates in an oscillatory fashion.Similar oscillations are also visible in hydrodynamic degrees of freedom through coupled dynamics.The oscillations are reduced when dissipation is present.We also found that the quark density,which initially peaked at the boundary of the boost invariant region,evolves toward forward rapidity with the peak velocity correlated with the velocity of longitudinal expansion.The peak broadens during this evolution.The corresponding chemical potential rises due to simultaneous decrease of density and temperature.We compared the cases with and without dissipation for the order parameter and also the standard hydrodynamics without order parameter.We found that the corresponding effects on temperature and chemical potential can be understood from the conservation laws and different speeds of equilibration of the order parameter in the three cases.     

Quintic Nonlinearity Induced Solitary Waves in Plasma Physics

The quintic nonlinearity is important in the study of the nonlinear interaction between Langmuir waves and electrons in plasma.Using the pseudoenergy approach,five types of solitary wave solution are obtained explicitly. Only one of these is the modification of the soliton of the cubic nonlinear Schrodinger equation and can be treated perturbatively.However,other four types of solitary wave solution are all induced by the quintic nonlinearity and cannot be treated perturbatively from the solutions of the cubic nonlinear Schrodinger equation.     

Fundamental modes in waveguide pipe twisted by saturated double-well potential

We study fundamental modes trapped in a rotating ring with a saturated nonlinear double-well potential. This model, which is based on the nonlinear Schrödinger equation, can be constructed in a twisted waveguide pipe in terms of light propagation, or in a Bose–Einstein condensate (BEC) loaded into a toroidal trap under a combination of a rotating π-out-of-phase linear potential and nonlinear pseudopotential induced by means of a rotating optical field and the Feshbach resonance. Three types of fundamental modes are identified in this model, one symmetric and the other two asymmetric. The shape and stability of the modes and the transitions between different modes are investigated in the first rotational Brillouin zone. A similar model used a Kerr medium to build its nonlinear potential, but we replace it with a saturated nonlinear medium. The model exhibits not only symmetry breaking, but also symmetry recovery. A specific type of unstable asymmetric mode is also found, and the evolution of the unstable asymmetric mode features Josephson oscillation between two linear wells. By considering the model as a configuration of a BEC system, the ground state mode is identified among these three types, which characterize a specific distribution of the BEC atoms around the trap.     

Optical solitons in PT periodic potentials

We investigate the effect of nonlinearity on beam dynamics in parity-time (PT) symmetric potentials. We show that a novel class of one- and two-dimensional nonlinear self-trapped modes can exist in optical PT synthetic lattices. These solitons are shown to be stable over a wide range of potential parameters. The transverse power flow within these complex solitons is also examined.     

亮孤子在宇称时间对称波导中的传输和控制

基于光波在宇称-时间(PT)对称波导中传输的理论模型, 数值研究了亮孤子在呈高斯分布的PT对称克尔非线性平板波导中的传输和控制. PT对称波导, 要求波导的折射率分布呈偶对称, 而增益/损耗分布呈奇对称. 结果表明: 当波导的折射率分布强度为正时, PT对称波导的中心折射率最大, 即使没有自聚焦克尔非线性效应, PT对称波导也可以束缚光波, 形成波浪形光束且长距离传输; 当折射率分布强度为负时, PT对称波导的中心折射率最小, 光波的传输方向发生偏移. 而增益/损耗分布可控制光波的偏移方向: 增益/损耗分布强度为正, 光波向左偏移; 强度为负, 光波向右偏移; 强度为零时, 光波被分为两束. 且当折射率分布强度为负时, 可以很好地抑制相邻亮孤子间的相互作用. 该研究结果可为未来PT对称波导在全光控制方面的应用提供一定的理论依据.     

Defect solitons supported by parity-time symmetric defects in superlattices

The existence and stability of defect solitons supported by parity-time (PT) symmetric defects in superlattices are investigated. In the semi-infinite gap, in-phase solitons are found to exist stably for positive defects, zero defects, and negative defects. In the first gap, out-of-phase solitons are stable for positive defects or zero defects, whereas in-phase solitons are stable for negative defects. For both the in-phase and out-of-phase solitons with the positive defect and in-phase solitons with negative defect in the first gap, there exists a cutoff point of the propagation constant below which the defect solitons vanish. The value of the cutoff point depends on the depth of defect and the imaginary parts of the PT symmetric defect potentials. The influence of the imaginary part of the PT symmetric defect potentials on soliton stability is revealed.     

The optical properties of two-dimensional Scarff parity–time symmetric potentials

We investigate the optical properties with two-dimensional (2D) Scarff parity–time (PT) symmetric potentials, including linear case, and self-focusing and self-defocussing nonlinear cases. For linear case, the PT-breaking points, the eigenvalues and eigenfunction for different modulated depths of 2D Scarff PT symmetry complex potential are obtained numerically. The PT-breaking points increase linearly with increasing the real part of the modulated depths of PT potential. Below the PT-breaking points, the eigenvalues of linear modes are real, however, eigenvalues of linear modes are complex above the PT-breaking points. For nonlinear cases, the existence of fundamental and multipole solitons is studied in self-focusing and self-defocussing media. The eigenvalue for linear case is equal to the critical propagation constant _bc$b_c$ of soliton existing.     

Nonlinear r-modes in rapidly rotating relativistic stars

The r-mode instability in rotating relativistic stars has been shown recently to have important astrophysical implications, provided that r-modes are not saturated at low amplitudes by nonlinear effects or by dissipative mechanisms. Here, we present the first study of nonlinear r-modes in isentropic, rapidly rotating relativistic stars, via 3D general-relativistic hydrodynamical evolutions. We find that (1) on dynamical time scales, there is no strong nonlinear coupling of r-modes to other modes at amplitudes of order one-the maximum r-mode amplitude is of order unity. (2) r-modes and inertial modes in isentropic stars are predominantly discrete modes. (3) The kinematical drift associated with r-modes appears to be present in our simulations, but confirmation requires more precise initial data.     

Soliton complexes and flat-top nonlinear modes in optical lattices

We describe a continuous analog of the quasirectangular flat-top nonlinear modes earlier found for discrete nonlinear models. We show that these novel nonlinear modes can be understood as multi-soliton complexes with either in-phase or out-of-phase neighboring solitons trapped by the periodic potential of the lattice. We demonstrate a link between the flat-top states and the truncated nonlinear Bloch waves, and discuss how these nonlinear localized modes can be monitored experimentally in photonics and Bose–Einstein condensates. PACS 42.65.Tg; 42.65.Jx; 03.75.Lm     

Bifurcated overtones of one-way localized Fabry-Pérot resonances in parity-time symmetric optical lattices

Since the first observation of parity-time(PT) symmetry in optics, varied interesting phenomena have been discovered in both theories and experiments, such as PT phase transition and unidirectional invisibility, which turns PT-symmetric optics into a hotspot in research. Here, we report on the one-way localized Fabry-Pérot(FP) resonance, where a welldesigned PT optical resonator may operate at exceptional points with bidirectional transparency but unidirectional field localization. Overtones of such one-way localized FP resonance can be classified into a blue shifted branch and a red shifted branch. Therefore, the fundamental resonant frequency is not the lowest one. We find that the spatial field distributions of the overtones at the same absolute order are almost the same, even though their frequencies are quite different.     

Effect of near ${ \mathcal P }{ \mathcal T }$-symmetric potentials on nonlinear modes for higher-order generalized Ginzburg–Landau model

In this paper, we study the higher-order generalized Ginzburg–Landau model which contributes to describing the propagation of optical solitons in fibers. By means of the Hirota bilinear method, the analytical solutions are obtained and the effect of relevant parameters is analyzed. Modulated by the near parity-time-symmetric potentials, the nonlinear modes with 5% initial random noise are numerically simulated to possess stable evolution. Furthermore, the evolution of nonlinear modes is displayed through the adiabatical change of some parameters. The investigation of the present work is intended as a contribution to the work for the higher-order generalized Ginzburg–Landau model.     

Analytical Study on Propagation Dynamics of Optical Beam in Parity-Time Symmetric Optical Couplers

We present exact analytical solutions to parity-time(P T) symmetric optical system describing light transport in P T-symmetric optical couplers. We show that light intensity oscillates periodically between two waveguides for unbroken P T-symmetric phase, whereas light always leaves the system from the waveguide experiencing gain when light is initially input at either waveguide experiencing gain or waveguide experiencing loss for broken P T-symmetric phase. These analytical results agree with the recent experimental observation reported by Ru¨ter et al. [Nat. Phys.6(2010) 192]. Besides, we present a scheme for manipulating P T symmetry by applying a periodic modulation. Our results provide an efficient way to control light propagation in periodically modulated P T-symmetric system by tuning the modulation amplitude and frequency.