Preface to the special topic: Nonlinear system theory and its frontier applications
探索非线性系统理论并将其应用于物理学的诸多前沿科学领域是当前物理研究的热点课题之一, 其发展为物理学新现象、新概念、新机理提供了基准, 并在物理学发展过程中起到了关键的基础支撑和重要的推动作用. 所有复杂非线性系统都由许多不同的相互作用部分组成, 几个世纪以来,物理学家一直在研究它们, 这些系统很难用数学方法来描述——它们可能被大量的因素影响或者受随机因素的支配. 2021 年的诺贝尔物理学奖授予了 Syukuro Manabe, Klaus Hasselmann 和Giorgio Parisi 三位物理学家, 表彰他们“对我们理解复杂系统的开创性贡献”. 可见, 非线性物理中的复杂系统在我们生活中扮演着重要角色. 虽然非线性系统领域蓬勃发展了几十年, 但是不断有新的非线性系统被发现, 不断有新的解析和数值方法出现. 特别地, 随着实验技术的不断进步, 具有非局域相互作用和李黄杨修正的各类广义非线性系统已在实验室里被成功制备, 这使得理论上的研究更具有现实意义. 目前, 此类研究正处于蓬勃发展的早期阶段, 还存在着许多亟待探索的新奇非线性现象和丰富的动力学行为, 在未来几年仍将是数学物理领域活跃的热点之一. 在这样一个大背景下, 对上述各类非线性系统进行更加深入的研究显得非常及时和必要.
为进一步促进国内同行的交流, 在《物理学报》编辑部的大力支持下, 我们邀请了国内活跃在该领域的部分科学家撰写了 18 篇相关论文, 其中 2 篇综述论文, 其他 16 篇为研究论文. 鉴于非线性物理属于交叉学科, 具有多样性及复杂性的特点, 本专题只能重点介绍其在冷原子物理、可积系统、非线性光学、深度学习等领域的部分研究成果, 内容涉及带隙孤子、多极矢量孤子、高阶怪波、非正则涡旋、淬火动力学、狄拉克磁单极势、非局域孤子、初值问题等典型的非线性拓扑激发及其动力学. 我们希望本专题能够尽可能反映该方向的研究现状, 为青年学者选择科研方向、确定研究课题以及从事相关领域研究的人员提供一点帮助, 促进我国在非线性系统理论及其前沿应用的发展.鉴于非线性物理在量子物理、凝聚态物理、非线性光学等领域中的广泛存在, 其带来的新奇物理现象无法一概而论. 同时, 受水平及时间所限, 本专题对其前沿应用介绍难免挂一漏万, 不足之处恳请各位同仁不吝指正.
2023, 72 (10): 100308.
doi: 10.7498/aps.72.20222195
Abstract +
The dynamical stability properties of surface gap solitons in quasi-one-dimensional Bose-Einstein condensate loaded in the interface between uniform media and a semi-infifinite Jacobian elliptic sine potential with three-body interactions are investigated numerically. Under the mean-fifield approximation, the dynamical behaviors can be well-described by the nonlinear cubic-quintic Gross-Pitaevskii equation. Firstly, many kinds of surface gap solitons, including the surface bright solitons, surface kink solitons and surface bubble solitons, are obtained numerically by the Newton-conjugate gradient method. The surface bright solitons can be excited in the gap only for the case that the chemical potential is negative and their power is beyond a threshold value. All of them are not bifurcated from the Bloch band. A class of surface solitons with new structures, named the surface dark solitons, can be formed when the three-body interactions are taken into account. The surface dark solitons can exist not only in gap but also in band. The numerical results indicate that the amplitude of the surface gap solitons decreases as the three-body interaction strength increases. Both linear stability analysis and nonlinear dynamical evolution methods are applied to investigate the stability properties of surface gap solitons. For surface bright solitons in the semi-infinite gap, there is a critical value when the chemical potential is given. The surface bright solitons become linearly stable as the three-body interaction exceeds the critical value, or they are linearly unstable. Therefore, the three-body interaction strength plays an important role on the stability of surface gap solitons. One can change the dynamical behaviors of surface gap solitons by adjusting the three-body interaction strength in experiments. Numerical results also show that both stable and unstable surface kink solitons exist. However, all the surface bubble solitons are unstable.
2023, 72 (10): 100201.
doi: 10.7498/aps.72.20230241
Abstract +
The Korteweg-de Vries (KdV) equation is a mathematical model that describes the propagation of long waves in dispersive media. It takes into account both nonlinearity and dispersion, and is particularly useful for modeling phenomena like solitons. The nonlinear Schrödinger (NLS) equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. It is a useful model for describing many physical systems, including Bose-Einstein condensates, optical fibers, and water waves. A system that couples the KdV and NLS equations can model the interaction of long and short waves. This system combines the strengths of both models. The long waves described by the KdV equation can affect the behavior of the short waves described by the NLS equation, while the short waves can in turn affect the behavior of the long waves. Such a coupled system has been studied extensively over the last few decades, and has led to important insights into many physical systems. This paper considers the existence of local solutions to the Cauchy problem of KdV-Schrödinger nonlinear system on the basis of literature (Bernard D, Nghiem V N, Benjamin L S 2016 J. Phys. A: Math. Theor. 49 415501 ), and also gives the existence space of the local solutions.
2023, 72 (10): 100504.
doi: 10.7498/aps.72.20222430
Abstract +
In the study of telecommunication system, the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation is used as the optical solitons transmission model, which not only explains the physical meaning of the existing model with quintic terms, but also has more nonlinear dynamics characteristics of the higher dimensional system than the lower dimensional system. In this paper, the analytical soliton solutions of the (3+1)-dimensional cubic-quintic CGL equations with variable coefficients are obtained by using the modified Hirota method. By selecting certain parameters of the nonlinear coefficients and spectral filtering terms, a special kind of mixed soliton solution is obtained, which has the characteristics of bright soliton, dark soliton and kinked soliton at the same time. Subsequently, the influence of changing the nonlinear, spectral filtering, linear loss parameters and other parameters on the transmission characteristics of solitons is discussed respectively, so as to realize the control of optical solitons, which can not only control the propagation of optical solitons in different forms, but also can realize the adjustment of the amplitude and pulse width of the pulse and control the propagation direction and energy of the pulse for the mixed solitons of a particular form. The research results of high dimensional CGL system in this paper can be applied to nonlinear optical system, ultra-fast optical digital logic system and other different experiments and application fields.
2023, 72 (10): 100203.
doi: 10.7498/aps.72.20230063
Abstract +
The paper is a review of the bilinearization-reduction method which provides an approach to obtain solutions to integrable systems. Many integrable coupled systems can be bilinearized and their solutions are presented in terms of double Wronskians (or double Casoratians in discrete case). The bilinearization-reduction method is based on bilinear equations and solutions in double Wronskian/Casoratian form. For those integrable equations that are reduced from coupled systems, one can first solve the unreduced coupled system, obtaining their solutions in double Wronskian/Casoratian form, then, implement suitable reduction techniques, so that solutions of the reduced equation can be obtained as reductions of those of the unreduced coupled system. The method proves effective in solving not only classical integrable equations but also the nonlocal ones. The so-called nonlocal integrable equations were introduced by Ablowitz and Musslimani via reductions with reverse-space (or reverse-time, or reverse-space-time). Note that this method particularly provides a convenient bilinear approach to solve nonlocal integrable systems. In this review, the nonlinear Schrödinger hierarchy and the differential-difference nonlinear Schrödinger equation are employed as demonstrative examples to elaborate this method. These two examples will be pedagogically helpful in understanding the reduction technique. The reduction is implemented by imposing suitable constraints on the basic column vectors of the double Wronskian/Casoratian. Realizations of the constraints are converted to solve a set of matrix equations which varies with the constraints. Special solutions of the matrix equations are provided, which are also helpful in understanding the eigenvalue structure of the involved spectral problems corresponding to the considered equations. Other examples include the Fokas-Lenells equation and the nonlinear Schrödinger equation with nontrivial background. Since many nonlinear equations with physical significance are integrable as reductions of integrable coupled systems, the paper provides a review as well as an introduction about the bilinearization-reduction method that can be used to solve these nonlinear integrable models.
2023, 72 (10): 100204.
doi: 10.7498/aps.72.20222418
Abstract +
The study of integrable systems is one of important topics both in physics and in mathematics. However, traditional studies on integrable systems are usually restricted in (1+1) and (2+1) dimensions. The main reasons come from the fact that high-dimensional integrable systems are extremely rare. Recently, we found that a large number of high dimensional integrable systems can be derived from low dimensional ones by means of a deformation algorithm. In this paper, the (1+1) dimensional Kaup-Newell (KN) system is extended to a (4+1) dimensional system with the help of the deformation algorithm. In addition to the original (1+1) dimensional KN system, the new system also contains three reciprocal forms of the (1+1) dimensional KN system. The model also contains a large number of new (D+1) dimensional ($D \leqslant 3$ ) integrable systems. The Lax integrability and symmetry integrability of the (4+1) dimensional KN system are also proved. It is very difficult to solve the new high-dimensional KN systems. In this paper, we only investigate the traveling wave solutions of a (2+1) dimensional reciprocal derivative nonlinear Schrödinger equation. The general envelope travelling wave can be expressed by a complicated elliptic integral. The single envelope dark (gray) soliton of the derivative nonlinear Schödinger equation can be implicitly written.
2023, 72 (10): 107502.
doi: 10.7498/aps.72.20230345
Abstract +
Inertia effect should be considered in ferromagnet magnetization dynamics on a sub picosecond-to-femtosecond-time scale. The inertia effect can be described by the inertial Landau-Lifshitz-Gilbert equation. This paper mainly introduces some theoretical and experimental developments of ultrafast ferromagnetic resonance, magnetization reversal and inertial spin dynamics. These results will be helpful in better understanding the basic mechanism of ultrafast demagnetization and magnetization reversal, and deepen the understanding of the microscopic mechanism of magnetic inertia. In the end, the development trend of future experimental and theoretical research are also presented.
2023, 72 (10): 100501.
doi: 10.7498/aps.72.20222416
Abstract +
In this paper, we deeply investigate the phase evolution and the underlying topological vector potential in the nonlinear interference of solitons. Based on the double-soliton solution of 1D nonlinear Schrödinger equation, we find that the density zeros of wave function generally exist in the extended complex space, each density zero corresponds to the vector potential produced by Dirac magnetic monopole. The vector potential field is composed of periodically distributed Dirac magnetic monopole pairs with opposite magnetic charges. By observing the motion of magnetic monopoles, we can conveniently understand the phase evolution characteristics during the interference process. In particular, we find that the collision of a pair of magnetic monopoles with opposite charge on the real axis corresponds exactly to the $ \pm\pi $ jump of the wave function phase at nodes. For comparison, we also discuss Dirac magnetic monopoles and vector potential field in linear wave packet interference case. The results show that the Dirac magnetic monopole potential widely exists in the interference phenomena of wave fields, and the distribution of magnetic monopoles in the extended complex space can be used to distinguish the topological properties behind the linear and nonlinear interference process.
2023, 72 (10): 100306.
doi: 10.7498/aps.72.20222289
Abstract +
Vortex excitations triggered by nonlinear interactions in Bose-Einstein condensates have attracted interest in the study of ultracold atoms. However, most studies focus on canonical vortex states with integer topological charges. In this paper, we study the dynamic properties of noncanonical vortex condensates with three phase distributions: power-exponent, new type power-exponent and oscillation type. The results show that the noncanonical vortices are dynamic unstable and their density distributions obviously depend on the phase parameters of the initial optical phase masks. Different noncanonical vortices decay into canonical clusters with diverse configurations showing rich topological excitation patterns. In particular, a new power exponential noncanonical vortex state decays into a stable canonical polygonal vortex cluster structure. Because the phase structures of the noncanonical optical vortices destroy the rotational symmetry of the condensate, the angular momentum of the condensate is no longer quantized, and its value changes with the power of the azimuthal angle of the optical field or the oscillation frequency, which is obviously different from the evolution of the corresponding noncanonical vortex optical field itself. In the dynamical process, the center-of-mass trajectory of noncanonical vortex condensates with the new type of power exponent phase is always a point, while for the noncanonical vortex condensates with power exponent and oscillating phase, the center-of-mass trajectories are ellipses centering at the origin of coordinates.
2023, 72 (10): 100503.
doi: 10.7498/aps.72.20230172
Abstract +
Since the Whitham modulation theory was first proposed in 1965, it has been widely concerned because of its superiority in studying dispersive fluid dynamics and dealing with discontinuous initial value problems. In this paper, the Whitham modulation theory of the defocusing nonlinear Schrödinger equation is developed, and the classification and evolution of the solutions of discontinuous initial value problem are studied. Moreover, the dispersive shock wave region, the rarefaction wave region, the unmodulated wave region and the plateau region are distinguished. Particularly, the correctness of the results is verified by direct numerical simulation. Specifically, the solutions of 0-phase and 1-phase and their corresponding Whitham equations are derived by the finite gap integration method. Also the Whitham equation of genus N corresponding to the N-phase periodic wave solution is derived. The basic structures of rarefaction wave and dispersive shock wave are given, in which the boundaries of the regions are calculated in detail. The Riemann invariants and density distributions of dispersive fluids in each case are discussed. When the initial value is fixed as a special one, the vacuum point is considered and analyzed in detail. In addition, the oscillating front and the soliton front in the dispersive shock wave are considered. In fact, the Whitham modulation theory has many wonderful applications in real physics and engineering. The dam problem is investigated as a special Riemann problem, the piston problem of dispersive fluid is analyzed, and the novel undular bores are found.
2023, 72 (10): 100202.
doi: 10.7498/aps.72.20222381
Abstract +
In recent years, physics-informed neural networks (PINNs) have attracted more and more attention for their ability to quickly obtain high-precision data-driven solutions with only a small amount of data. However, although this model has good results in some nonlinear problems, it still has some shortcomings. For example, the unbalanced back-propagation gradient calculation results in the intense oscillation of the gradient value during the model training, which is easy to lead to the instability of the prediction accuracy. Based on this, we propose a gradient-optimized physics-informed neural networks (GOPINNs) model in this paper, which proposes a new neural network structure and balances the interaction between different terms in the loss function during model training through gradient statistics, so as to make the new proposed network structure more robust to gradient fluctuations. In this paper, taking Camassa-Holm (CH) equation and DNLS equation as examples, GOPINNs is used to simulate the peakon solution of CH equation, the rational wave solution of DNLS equation and the rogue wave solution of DNLS equation. The numerical results show that the GOPINNs can effectively smooth the gradient of the loss function in the calculation process, and obtain a higher precision solution than the original PINNs. In conclusion, our work provides new insights for optimizing the learning performance of neural networks, and saves more than one third of the time in simulating the complex CH equation and the DNLS equation, and improves the prediction accuracy by nearly ten times.
2023, 72 (10): 104202.
doi: 10.7498/aps.72.20230096
Abstract +
Realizing stable high-dimensional light solitons is a long-standing goal in the study of nonlinear optical physics. However, in high-dimensional space, the light field will inevitably be distorted due to diffraction. In order to solve the diffraction effect in nonlinear Kerr media and achieve the spatial localization of light fields, we propose a scheme to generate stable two-dimensional (2D) solitons in a cold Rydberg atomic system with a Bessel optical lattice, where a three-level atomic structure, a weak probe laser field, and a strong control field constitute the Rydberg-dressed atomic system. When the local nonlinearity, Bessel potential, and nonlocal nonlinearity which is caused by the long-range Rydberg-Rydberg interaction (RRI) between Rydberg atoms are balanced, the probe field can be localized. Under the approximation of electric dipole and rotating wave, the stable solution of probe field is obtained by solving Maxwell-Bloch equations numerically. A cluster of 2D spatial solitons, including fundamental, two-pole, quadrupole and vortex solitons, is found in this system. Among them, the fundamental, dipole and quadrupole have, one, two, and four intensity centers, respectively. Vortex solitons, on the other hand, exhibit vertical characters in profiles and phase structures. The formation and transmission of these solitons can be controlled by system parameters, such as the propagation coefficient, the degree of nonlocal nonlinearity, and Bessel lattice strength. The stable regions of these solitons are determined by anti Vakhitov Kolokolov (anti-VK) criterion and linear stability analysis method. It is found that four kinds of solitons can be generated and stably propagate in space with proper parameters. Owing to the different structures of the poles, the fundamental state and vortex state remain stable, while the quadrupole ones are unstable. In the modulation of solitons, there is a cutoff value of propagation constant ${b_{{\text{co}}}}$ , only below which value, the solitons can propagate stably. The light intensity of soliton shows a periodic behavior by tuning Bessel lattice strength. The period of the intensity decreases with the order of the solitons as a result of the interaction between the poles. It is also found that the solitons are more stable with weak nonlocal nonlinearity coefficient. This study provides a new idea for the generation and regulation of optical solitons in high dimensional space.
2023, 72 (10): 100307.
doi: 10.7498/aps.72.20222292
Abstract +
The experimental realization of Rydberg dressing and spin-orbit coupling greatly broadens the research field of ultracold atoms as a quantum simulation platform. Very recently, moiré lattices have attracted intensive study, ranging from condensed matter to ultracold physics. In this paper, the ground-state structure of Rydberg-dressed Bose gas with spin-orbit coupling and confined in moiré lattices is studied, and the effects of nonlocal Rydberg interaction and spin-orbit coupling on the ground state of the system are explored. Our results show that the system has no translational symmetry due to the presence of nonlocal Rydberg interaction, and more and more regular periodic structures present with the increases of the strength of nonlocal Rydberg interaction. In the presence of spin-orbit coupling, the Hamiltonian of the system has an imaginary part, and the phase of the system is not uniformly distributed. It is found that the ground state of the system with spin-orbit coupling present more abundant internal structure base on these periodic structures. The results pave the way for future study of moiré physics in ultracold atom system.
2023, 72 (10): 106701.
doi: 10.7498/aps.72.20222319
Abstract +
In a quantum system with spin, spin-orbit coupling is manifested by linking the spin angular momentum of a particle with its orbital angular momentum, which leads to many exotic phenomena. The experimental realization of synthetic spin-orbit coupling effects in ultra-cold atomic systems provides an entirely new platform for exploring quantum simulations. In a spinor Bose-Einstein condensate, the spin-orbit coupling can change the properties of the system significantly, which offers an excellent opportunity to investigate the influence of spin-orbit coupling on the quantum state at the macroscopic level. As typical states of macroscopic quantum effects, solitons in spin-orbit coupled Bose-Einstein condensates can be manipulated by spin-orbit coupling directly, which makes the study on spin-orbit coupled Bose-Einstein condensates become one of the hottest topics in the research of ultracold atomic physics in recent years. This paper investigates exact vector soliton solutions of the Gross-Pitaevskii equation for the one-dimensional spin-orbit coupled binary Bose-Einstein condensates, which has four parameters $\mu$ , $\delta$ , $\alpha$ and $\beta$ , where $\mu$ denotes the strength of the spin-orbit coupling, $\delta$ is the detuning parameter, $\alpha$ and $\beta$ are the parameters of the self- and cross-interaction, respectively. For the case $\beta=\alpha$ , by a direct ansatz, two kinds of stripe solitons, namely, the oscillating dark-dark solitons are obtained; meanwhile, a transformation is presented such that from the solutions of the integrable Manakov system, one can get soliton solutions for the spin-orbit coupled Gross-Pitaevskii equation. For the case $\beta=3\alpha$ , a bright-W type soliton for $\alpha>0$ and a kink-antikink type soliton for $\alpha<0$ are presented. It is found that the relation between $\mu$ and $\delta$ can affect the states of the solitons. Based on these solutions, the corresponding dynamics and the impact of the spin-orbit coupling effects on the quantum magnetization and spin-polarized domains are discussed. Our results show that spin-orbit coupling can result in rich kinds of soliton states in the two-component Bose gases, including the stripe solitons as well as the classical non-stripe solitons, and various kinds of multi-solitons. Furthermore, spin-orbit coupling has a remarkable influence on the behaviors of quantum magnetization. In the experiments of Bose-Einstein condensates, there have been many different methods to observe the soliton states of the population distribution, the magnetic solitons, and the spin domains, so our results provide some possible options for the related experiments.
2023, 72 (10): 100505.
doi: 10.7498/aps.72.20230425
Abstract +
In non-conservative nonlinear systems, the basic physical mechanics of soliton generation is that the kinetic energy and nonlinear terms of the system, as well as the gain and dissipation terms reach a double dynamic balance. How to generate stable free high-dimensional solitons in such a system is currently a challenging topic in soliton theory. In this article, we propose a theoretical scheme for realizing two-dimensional free bright solitons in exciton-polariton Bose-Einstein condensates, which proposes a physical mechanism for generating stable two-dimensional free space bright solitons through time periodic modulation interactions and a dual balance between gain and dissipation. In this end, firstly, we obtain the dynamic equations of two-dimensional bright soliton parameters through the Lagrange variational method, and obtain its dynamically stable parameter space. Secondly, the evolution of the generalized dissipative Gross-Pitaveskii equation is numerically simulated to verify the stability of two-dimensional bright solitons. Finally, we add Gaussian noise to simulate a real experimental environment and find that two-dimensional bright solitons are also stable within the observable time range of the experiment. Our experimental scheme opens the door to the study of bright solitons in high-dimensional free space in non-conservative systems.
2023, 72 (10): 100502.
doi: 10.7498/aps.72.20222284
Abstract +
We construct the coupled self-defocusing saturated nonlinear Schrödinger equation and obtain the dipole-dipole, tripole-dipole and dipole-tripole vector soliton solutions by changing the potential function parameters and using the square operator method of power conservation. With the increase of soliton power, the dipole-dipole, tripole-dipole and dipole-tripole vector solitons can all exist. The existence of the three kinds of vector solitons is obviously modulated by the potential function. The existence domain of three kinds of vector solitons, modulated by the potential function, is given in this work. The stability domains of three vector solitons are modulated by the soliton power of each component. The stability regions of three kinds of vector solitons expand with the increase of the power of two-component soliton. With the increase of saturation nonlinear strength, the power values of the tripole-dipole and dipole-tripole vector solitons at the critical points from stable state to unstable state decrease gradually, and yet the power of the soliton at the critical point from the stable state to the unstable state does not change.
2023, 72 (10): 104204.
doi: 10.7498/aps.72.20222298
Abstract +
General higher-order rogue wave solutions to the space-shifted $\mathcal{PT}$ -symmetric nonlocal nonlinear Schrödinger equation are constructed by employing the Kadomtsev-Petviashvili hierarchy reduction method. The analytical expressions for rogue wave solutions of any Nth-order are given through Schur polynomials. We first analyze the dynamics of the first-order rogue waves, and find that the maximum amplitude of the rogue waves can reach any height larger than three times of the constant background amplitude. The effects of the space-shifted factor $x_0$ of the $\mathcal{PT}$ -symmetric nonlocal nonlinear Schrödinger equation in the first-order rogue wave solutions are studied, which only changes the center positions of the rogue waves. The dynamical behaviours and patterns of the second-order rogue waves are also analytically investigated. Then the relationships between Nth-order rogue wave patterns and the parameters in the analytical expressions of the rogue wave solutions are given, and the several different patterns of the higher-order rogue waves are further shown.
2023, 72 (10): 100309.
doi: 10.7498/aps.72.20222401
Abstract +
In this work, we study the non-equilibrium quench dynamics from the superfluid stripe phase to the supersolid phase of a two dimensional spin-orbital coupled interacting Bose-Einstein condensate in the presence of a one dimensional optical lattice. The quench protocol here is constructed through varying the lattice depth linearly with the evolution time. By using the time-dependent Gutzwiller method, various physical quantities, such as the vortex number and the overlap of wave-function, have been investigated with respect to the quench time. Through analyzing the dynamical behavior of the above physical quantities, we find out the transition time of the quench procedure, which captures the freeze out time indicating the moment that the system catches the quench speed beginning to evolve quickly. Before the transition time, the dynamics is frozen and the state of the system cannot follow the changes in the Hamiltonian. While passing the transition time, we find that there are significant alterations to both the vortex number and the wave-function. At the transition time, on one hand the vortex number abruptly increases from zero; on the other hand the overlap of wave-function departures from 1 shortly. These signatures indicate that the system evolves rapidly when passing the transition time. Furthermore, we also find that due to the presence of spin-orbital coupling, the spin texture represents a periodic magnetic structure accompanying with the emergence of the supersolid dynamically. It is shown that during the quench procedure, the density distribution of the system are always accompanied with the spatial structure of spin texture, i.e., the central position of topological spin skyrmion (antiskyrmion) corresponding to the minimum position of the density distribution. The topological charge of the above spin structures also shows interesting dynamical properties. We find that the quantized topological charge appears with the emergence of the supersolid dynamically.
2023, 72 (10): 104205.
doi: 10.7498/aps.72.20230082
Abstract +
Parity-time (PT) symmetric is not a necessary condition for achieving a real spectrum and some studies about realizing real spectra in non-PT-symmetric systems with arbitrary gain–loss profiles have been presented recently. By tuning the free parameters in non-PT-symmetric potentials, phase transition could also be induced. Above phase transition point, discrete complex eigenvalues bifurcate out from continuous real eigenvalues in the interior of the continuous spectrum. In this work, we investgate the existence and stability of solitons in nonlocal nonlinear couplers with non-PT-symmetric complex potentials both below and above phase transition. There are several discrete eigenvalues in the linear spectra of the non-PT-symmetric system used here. With the square-operator iteration method, we find that different continuous families of solitions can bifurcate from different discrete linear eigenvalues. Moreover, linear-stability analysis collaborated with direct numerical propagation simulations demonstrates that the nonlocal solitions can be stable in a range of parameter values. we first address the cases below the phase transition. To be specific, when we fix the coupling coefficient and vary the degree of nonlocality, it’s found that fundamental solitons, dipole solitons, tripolar solitons, quadrupole solitons bifurcate from the largest,the second-largest, the third-largest and the fifth-largest discrete eigenvalue, respectively. These nonlocal solitons are all stable in the low power region. With an increase of the degree of nonlocality, the stability region shrinks for the fundamental solitons while it widens for the dipole and multiplole solitons. At the same time, the power of all the stable solitons increases with the increase of the degree of nonlocality. By varying the coupling coefficient, the arrangement of soliton families emerging in the discrete interval of the linear spectrum can be changed. For example, the dipole solitons bifurcate from the third-or fourth-largest discrete eigenvalue while the tripolar solitons bifurcate from the fifth largest discrete eigenvalue. Above phase transition,the fundamental solitons are unstable in the low and high power region but are stable in the moderate power region. The stability region shrinks with the increasing degree of nonlocality. We also find the family of dipole solitons bifurcates from the second-largest discrete eigenvalue, but all the dipole solitons are unstable. In addition, we find that the eigenvalues in linear-stability spectra of solitons emerge as conjugation pairs.
2023, 72 (18): 180304.
doi: 10.7498/aps.72.20231076
Abstract +
We investigate the dynamics of the plane wave state in one-dimensional spin-tensor-momentum coupled Bose-Einstein condensate. By using the Gaussian variational approximation, we first derive the equations of motion for the variational parameters, including the center-of-mass coordinate, momentum, amplitude, width, chirp, and relative phase. These variational parameters are coupled together nonlinearly by the spin-tensor-momentum coupling, Raman coupling, and the spin-dependent atomic interaction. By minimizing the energy with respect to the variational parameters, we find that the ground state is a biaxial nematic state, the momentum of the ground state decreases monotonically with the increase of the strength of the Raman coupling, and the parity of real part of the ground-state wave function is opposite to that of the imaginary part. The linear stability analysis shows that the ground state is dynamically stable under a perturbation, and exhibits three different oscillation excitation modes, the frequencies of which are related to the strength of the Raman coupling, the aspect ratio of the harmonic trap, and the strength of the atomic interaction. By solving the equations of motion for the variational parameters, we find that the system displays periodical oscillation in the dynamical evolution. These variational results are also confirmed by the direct numerical simulations of the Gross-Pitaevskii equations, and these findings reveal the unique properties given by the spin-tensor-momentum coupling.
