Solitons, nonlinear particle-like excitations with inalterable properties (amplitude, shape, and velocity) as they propagate, are omnipresent in many branches of science---and in physics in particular. Flat-top solitons are a novel type of bright solitons that have not been well explored in pure nonlinear media. Here, a model of nonlinear Kerr (cubic) media of ultracold atoms with spatially modulated repulsive interactions is proposed and shown to support a vast variety of stable flat-top matter-wave solitons, including one-dimensional (1D) flat-top fundamental and multipole solitons, two-dimensional (2D) flat-top fundamental and vortex solitons. We demonstrate that by varying the relevant physical parameters (nonlinearity coefficient and chemical potential) the ordinary bright (gaussian) solitons can transform into the novel flat-top solitons. The (in-)stability domains of the flat-top soliton families are checked by means of linear stability analysis and reconfirmed by direct numerical simulations. This model is generic in the contexts of nonlinear optics and Bose-Einstein condensates, which provide direct experimental access to observe the predicted solutions.
Spatially-periodic patterns are studied in nonlocally coupled Gross-Pitaevskii equation. We show first that spatially periodic patterns appear in a model with the dipole-dipole interaction. Next, we study a model with a finite-range coupling, and show that a repulsively coupled system is closely related with an attractively coupled system and its soliton solution becomes a building block of the spatially-periodic structure. That is, the spatially-periodic structure can be interpreted as a soliton lattice. An approximate form of the soliton is given by a variational method. Furthermore, the eff…
The existence and stability of stable bright solitons in one-dimensional (1D) media with a spatially periodical modulated Kerr nonlinearity are demonstrated by means of the linear-stability analysis and in direct numerical simulations. The nonlinear potential landscape can balance the fractional-order diffraction and thus stabilizes the solitons, making the model unique and governed by the recently introduced fractional Schrödinger equation with a self-focusing cubic nonlinear lattice. Both 1D fundamental and multihump solitons (in forms of dipole and tripole ones) are found, which occupy one …
The paper is devoted to numerical study of stability of nonlinear localized modes ("gap solitons") for the spatially one-dimensional Gross-Pitaevskii equation (1D GPE) with periodic potential and repulsive interparticle interactions. We use the Evans function approach combined with the exterior algebra formulation in order to detect and describe weak oscillatory instabilities. We show that the simplest ("fundamental") gap solitons in the first and in the second spectral gaps can undergo oscillatory instabilities for certain values of the frequency parameter (i.e., the chemical potential). The …
We demonstrate numerically the possibility of generation of vector platicons having orthogonally polarized components by the modulated pump at normal GVD. Dynamics of this process can be controlled by pump polarization and depends on both the spectral interval between pumped modes and the difference of their FSRs. We also show the possibility of the excitation of the hybrid soliton-platicon complexes if pumped modes have alternating signs of the GVD coefficient.
Solitons and Vortices in Two-dimensional Discrete Nonlinear Schrodinger Systems with Spatially Modulated Nonlinearity
我们考虑最近提出的模型[Phys. Rev. E 88，032905（2013）]的二维（2D）推广，其产生由散焦非线性支持的明亮离散孤子，散焦非线性的局部强度从中心到外围增长。我们从耦合消失的反连续（AC）极限出发，探索二维模型。在这个极限下，我们可以构造出各种各样的解，不仅包括单点激发，而且包括偶极和四极激发。此外，两个独立的家庭的解决方案进行了探讨：通常的“扩展”unstaggered亮孤子，其中所有的网站都在AC限制激发，具有相同的符号在整个晶格（他们代表最强大的状态支持的晶格，他们的1D对应物被认为是在上述工作中的1D亮孤子），和涡交叉，这是特定于2D设置。对于所有存在的状态，我们探索它们的稳定性（尽可能地分析）。通过直接模拟，展示了典型的不稳定性发展的情况。
We develop stability analysis for matter-wave solitons in a two-dimensional (2D) Bose-Einstein condensate loaded in an optical lattice (OL), to which periodic time modulation is applied, in different forms. The stability is studied by dint of the variational approximation and systematic simulations. For solitons in the semi-infinite gap, well-defined stability patterns are produced under the action of the attractive nonlinearity, clearly exhibiting the presence of resonance frequencies. The analysis is reported for several time-modulation formats, including the case of in-phase modulations of …
An integrable model possessing inhomogeneous ground states is proposed as an effective model of non-uniform quantum condensates such as supersolids and Fulde--Ferrell--Larkin--Ovchinnikov superfluids. The model is a higher-order analog of the nonlinear Schrödinger equation. We derive an $n$-soliton solution via the inverse scattering theory with elliptic-functional background, and reveal various kinds of soliton dynamics such as dark soliton billiards, dislocations, gray solitons, and envelope solitons. We also provide the exact bosonic and fermionic quasiparticle eigenstates and clarify their…
We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrodinger equation with a nonlinear lattice pseudopotential, i.e., periodically modulated coefficient in front of the cubic term, which takes both positive and negative local values. This model finds direct implementations in atomic Bose-Einstein condensates and nonlinear optics. The most essential finding is the existence of two branches of dipole solitons (DSs), which feature an antisymmetric shape, essentially squeezed into a single cel…
We obtain multivalley dark soliton solutions with asymmetric or symmetric profiles in multicomponent repulsive Bose-Einstein condensates by developing the Darboux transformation method. We demonstrate that the width-dependent parameters of solitons significantly affect the velocity ranges and phase jump regions of multivalley dark solitons, in sharp contrast to scalar dark solitons. For double-valley dark solitons, we find that the phase jump is in the range $[0,2π]$, which is quite different from that of the usual single-valley dark soliton. Based on our results, we argue that the phase jump …
We report theoretical prediction of exact localized solutions for dynamics of surface gravity waves, at the critical point kh=1.363, modelled by higher-order nonlinear Schrodinger equation. The model possess domain walls (kink solitons) and dark solitons modulated through different phase profiles. The parametric domains are delineated for the existence of soliton solutions. The effect of wave parameters have been discussed on the amplitude of surface gravity waves. Our work is motivated by Tsitoura et al. [1], on experimental and analytical observation of phase domain walls for deep water surf…
Competing nonlinearities, such as the cubic (Kerr) and quintic nonlinear terms whose strengths are of opposite signs (the coefficients in front of the nonlinearities), exist in various physical media (in particular, in optical and matter-wave media). A benign competition between self-focusing cubic and self-defocusing quintic nonlinear nonlinearities (known as cubic-quintic model) plays an important role in creating and stabilizing the self-trapping of D-dimensional localized structures, in the contexts of standard nonlinear Schrödinger equation. We incorporate an external periodic potential (…
The paper is devoted to nonlinear localized modes ("gap solitons") for the spatially one-dimensional Gross-Pitaevskii equation (1D GPE) with a periodic potential and repulsive interparticle interactions. It has been recently shown (G. L. Alfimov, A. I. Avramenko, Physica D, 254, 29 (2013)) that under certain conditions all the stationary modes for the 1D GPE can be coded by bi-infinite sequences of symbols of some finite alphabet (called "codes" of the solutions). We present and justify a numerical method which allows to reconstruct the profile of a localized mode by its code. As an example, t…
We characterize the soliton solutions and their interactions for a system of coupled evolution equations of nonlinear Schrödinger (NLS) type that models the dynamics in one-dimensional repulsive Bose-Einstein condensates with spin one, taking advantage of the representation of such model as a special reduction of a 2 x 2 matrix NLS system. Specifically, we study in detail the case in which solutions tend to a non-zero background at space infinities. First we derive a compact representation for the multi-soliton solutions in the system using the Inverse Scattering Transform (IST). We introduce …
The Hong-Ou-Mandel experiment leads indistinguishable photons simultaneously reaching a 50:50 beam splitter to emerge on the same port through two-photon interference. Motivated by this phenomenon, we consider numerical experiments of the same flavor for classical, {\it wave} objects in the setting of repulsive condensates. We examine dark solitons interacting with a repulsive barrier, a case in which we find no significant asymmetries in the emerging waves after the collision, presumably due to their topological nature. We also consider case examples of two-component systems, where the dark s…
Multi-ring necklace vortex solitons in Kerr nonlinear media with azimuthally modulated Bessel potentials
College of Physics and Electronics Engineering, Shanxi University（山西大学物理电子工程学院）；Shanxi Key Laboratory of Wireless Communication and Detection（山西省无线通信与探测重点实验室）；Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University（特拉维夫大学电气工程学院物理电子学系）；Instituto de Alta Investigación, Universidad de Tarapacá（塔拉帕卡大学高等研究院）
We elaborate a model of the interferometer which, unlike previously studied ones, uses a local (delta-functional) nonlinear repulsive potential, embedded into a harmonic-oscillator trapping potential, as the splitter for the incident soliton. An estimate demonstrates that this setting may be implemented by means of the localized Feshbach resonance controlled by a focused laser beam. The same system may be realized as a nonlinear waveguide in optics. Subsequent analysis produces an exact solution for scattering of a plane wave in the linear medium on the delta-functional nonlinear repulsive pot…
Solitons supported by singular spatial modulation of the Kerr nonlinearity
我们引入了一种基于一维非线性薛定谔方程（NLSE）的设置，其中自聚焦（SF）立方项由坐标的奇异函数|x| ^{-a}调制。它还可以与均匀自散焦（SDF）非线性背景相结合，并与类似的奇异排斥线性势相结合。该装置可以在光学和BEC中实现，旨在扩展NLSE中孤子的存在性和稳定性的一般分析。基本孤子的结果得到了解析和数值验证。孤子具有准尖点形状，二阶导数在中心发散，并且在0 < a < 1的整个存在范围内是稳定的。偶极（奇）孤子也被发现。它们在无限域内是不稳定的，但在半无限域内是稳定的。在SDF背景的存在下，存在两个基本孤子的子族，一个稳定的和一个不稳定的，它们一起存在于一个阈值以上的范数（孤子的总功率）。该系统还包括奇异排斥线性势，模拟分数维0 < D < 1的均匀空间中的孤子。一个二维扩展的系统，二次非线性的基础上，制定太。
We investigate numerically and theoretically the conditions leading to soliton ejection stimulated through the scattering of bright solitons by modulated reflectionless potential wells. Such potential wells allow for the possibility of controlled ejection of solitons with significantly high speeds. At the outset, we describe the scattering setup and characterise the soliton ejection in terms of the different parameters of the system. Then, we formulate a theoretical model revealing the underlying physics of soliton ejection. The model is based on energy and norm exchange between the incident s…
We address dissipative soliton formation in modulated PT-symmetric continuous waveguide arrays composed from waveguides with amplifying and absorbing sections, whose density gradually increases (due to decreasing waveguide separation) either towards the center of the array or towards its edges. In such a structure the level of gain/loss at which PT-symmetry gets broken depends on the direction of increase of the waveguide density. Breakup of the PT-symmetry occurs when eigenvalues of modes localized in the region, where waveguide density is largest, collide and move into complex plane. In this…
We consider two-dimensional spin-orbit coupled atomic Bose-Einstein condensate in a radially-periodic potential. The system supports different types of stable self-sustained states including radially-symmetric vorticity-carrying modes with different topological charges in two spinor components that may have multiring profiles and at the same time remain remarkably stable for repulsive interactions. Solitons of the second type show persistent rotation with constant angular frequency. They can be stable for both repulsive and attractive interatomic interactions. Due to inequivalence between cloc…
The real spectrum of bound states produced by PT-symmetric Hamiltonians usually suffers breakup at a critical value of the strength of gain-loss terms, i.e., imaginary part of the complex potential. On the other hand, it is known that the PT-symmetry can be made unbreakable in a one-dimensional (1D) model with self-defocusing nonlinearity whose strength grows fast enough from the center to periphery. The model is nonlinearizable, i.e., it does not have a linear spectrum, while the (unbreakable) PT symmetry in it is defined by spectra of continuous families of nonlinear self-trapped states (sol…
The wave properties of solitons in a two-component Bose-Einstein condensates with attractive interactions or repulsive interactions are investigated in detail. We demonstrate that dark solitons in one of component admit interference and tunneling behaviour, in sharp contrast to the scalar dark solitons and vector dark solitons. The analytic analysis of interference properties shows that spatial interference pattern is determined by the relative velocity of solitons, while temporal interference pattern depends on the velocities and widths of two solitons, differing from the interference propert…
The work is devoted to numerical investigation of stability of stationary localized modes ("gap solitons") for the one-dimentional nonlinear Schrödinger equation (NLSE) with periodic potential and repulsive nonlinearity. Two classes of the modes are considered: a bound state of a pair of in-phase and out-of-phase fundamental gap solitons (FGSs) from the first bandgap separated by various number of empty potential wells. Using the standard framework of linear stability analysis, we computed the linear spectra for the gap solitons by means of the Fourier collocation method and the Evans function…
We consider two-dimensional waveguide arrays with anisotropic coupling coefficients. We show using numerical and variational calculations that four stationary soliton types exist: Site-Centered, Bond-Centered, Hybrid-X and Hybrid-Y. For the isotropic case the last two modes become identical and equivalent to the known hybrid soliton. With a variational calculation using a gaussian trial function and six variational parameters corresponding to the soliton's position, width, and velocity components, the four stationary soliton types are reproduced and their equilibrium widths are accounted for a…
