Spatiotemporal self-similar solutions for the nonautonomous (3+1)-dimensional cubic-quintic Gross-Pitaevskii equation

With the help of similarity transformation, we obtain analytical spatiotemporal self-similar solutions of the nonautonomous （3＋1）-dimensional cubic-quintic Gross-Pitaevskii equation with time-dependent diffraction, nonlinearity, harmonic potential and gain or loss when two constraints are satisfied. These constraints between the system parameters hint that self-similar solutions form and transmit stably without the distortion of shape based on the exact balance between the diffraction, nonlinearity and the gain/loss. Based on these analytical results, we investigate the dynamic behaviours in a periodic distributed amplification system.

Exact analytical solutions of three-dimensional Gross-Pitaevskii equation with time-space modulation

By the generalized sub-equation expansion method and symbolic computation, this paper investigates the （3＋1）dimensional Gross-Pitaevskii equation with time-and space-dependent potential, time-dependent nonlinearity, and gain or loss. As a result, rich exact analytical solutions are obtained, which include bright and dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic function solutions. With computer simulation, the main evolution features of some of these solutions are shown by some figures. Nonlinear dynamics of a soliton pulse is also investigated under the different regimes of soliton management.

Application of the homotopy analysis method for the Gross-Pitaevskii equation with a harmonic trap

The Homotopy analysis method （HAM） is adopted to find the approximate analytical solutions of the Gross- Pitaevskii equation, a nonlinear Schrodinger equation is used in simulation of Bose-Einstein condensates trapped in a harmonic potential. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they fit very well with each other when the atomic interaction is weak.

Envelope Periodic Solutions to One-Dimensional Gross-Pitaevskii Equation in Bose-Einstein Condensation

In this paper, applying the dependent and independent variables transformations as well as the Jacobi elliptic function expansion method, the envelope periodic solutions to one-dimensional Gross-Pitaevskii equation in Bose-Einstein condensates are obtained.

A Spectral Integral Equation Solution of the Gross-Pitaevskii Equation

The Gross-Pitaevskii equation (GPE), that describes the wave function of a number of coherent Bose particles contained in a trap, contains the cube of the normalized wave function, times a factor proportional to the number of coherent atoms. The square of the wave function, times the above mentioned factor, is defined as the Hartree potential. A method implemented here for the numerical solution of the GPE consists in obtaining the Hartree potential iteratively, starting with the Thomas Fermi approximation to this potential. The energy eigenvalues and the corresponding wave functions for each successive potential are obtained by a spectral method described previously. After approximately 35 iterations a stability of eight significant figures for the energy eigenvalues is obtained. This method has the advantage of being physically intuitive, and could be extended to the calculation of a shell-model potential in nuclear physics, once the Pauli exclusion principle is allowed for.

Impact of Quantum Fluctuations on the Modulational Instability of a Modified Gross-Pitaevskii Equation with Two-Body Interaction

Modulational instability conditions for the generation of localized structures in the context of matter waves in Bose-Einstein condensates are investigated analytically and numerically. The model is based on a modified Gross-Pitaevskii equation, which account for the energy dependence of the two-body scattering amplitude. It is shown that the modified term due to the quantum fluctuations modify significantly the modulational instability gain. Direct numerical simulations of the full modified Gross-Pitaevskii equation are performed, and it is found that the modulated plane wave evolves into a train of pulses, which is destroyed at longer times due to the effects of quantum fluctuations.

Covariant Prolongation Structure, Conservation Laws and Soliton Solutions of the Gross-Pitaevskii Equation in the Bose-Einstein Condensate

In this paper, we investigate the Gross-Pitaevskii (GP) equation which describes the propagation of an electron plasma wave packet with a large wavelength and small amplitude in a medium with a parabolic density and constant interactional damping by the Covariant Prolongation Structure Theory. As a result, we obtain general forms of Lax-Pair representations. In addition, some hidden structural symmetries that govern the dynamics of the GP equation such as SL(2,R), SL(2,C), Virasoro algebra, SU(1,1) and SU(2) are unearthed. Using the Riccati form of the linear eigenvalue problem, infinite number of conservation laws of the GP equation is explicitly constructed and the exact analytical soliton solutions are obtained by employing the simple and straightforward Hirota’s bilinear method.

A High-Order Conservative Numerical Method for Gross-Pitaevskii Equation with Time-Varying Coefficients in Modeling BEC

We propose a high-order conservative method for the nonlinear Sehodinger/Gross-Pitaevskii equation with time- varying coefficients in modeling Bose Einstein condensation （BEC）. This scheme combined with the sixth-order compact finite difference method and the fourth-order average vector field method, finely describes the condensate wave function and physical characteristics in some small potential wells. Numerical experiments are presented to demonstrate that our numerical scheme is efficient by the comparison with the Fourier pseudo-spectral method. Moreover, it preserves several conservation laws well and even exactly under some specific conditions.

GP-STABILITY OF ROSENBROCK METHODS FOR SYSTEM OF DELAY DIFFERENTIAL EQUATION

The stability analysis of the Rosenbrock method for the numerical solutions of system of delay differential equations was studied. The stability behavior of Rosenbrock method was analyzed for the solutions of linear test equation. The result that the Rosenbrock method is GP-stable if and only if it is A-stable is obtained.

二维阻尼Gross-Pitaevskii(GP)方程的整体解

考察一类描述吸引Bose Einstein凝聚(BEC)的二维阻尼Gross Pitaevskii(GP)方程iφt=-Δφ+｜x｜2φ-｜φ｜2φ+ia｜φ｜4φ,t≥0,x∈R2,a<0.借鉴文献(CommunMathPhys,1983,87:567～576.)关于经典Schr dinger方程研究的思想和结果,建立GP方程与一个经典的非线性数量场方程的对应关系,得到方程的整体解存在的一个充分条件.

具阻尼的Gross-Pitaevskii(GP)方程的整体解(英文)

考虑临界的具阻尼的Gross Pitaevskii(GP)方程iφt=-Δφ+｜x｜2φ+g｜φ｜4/Dφ+iaφ,t≥0,x∈RD,g<0,a<0,这里D是空间维数.这个方程很好地描述了吸引的玻色 爱因斯坦凝聚(BEC).通过偏微分方程的严格理论和变分方法,获得了整体解的一个充分条件,而这个条件利用了非线性数量场方程-Δu+2bu-｜u｜4/Du=0的唯一正解.

具阻尼的Gross-Pitaevskii(GP)方程在二维空间中的稳定性

本文讨论出现在吸引玻色-爱因斯坦凝聚中的一类带调和势的阻尼非线性Schr dinger方程.对照玻色-爱因斯坦凝聚的物理性质,证明了阻尼参数存在一个门槛值,即当阻尼参数大于该门槛值时,初值问题的解整体存在;当阻尼参数小于该门槛值时,其初值问题的解将在有限时间内坍塌.

Analytical Solitary Wave Solutions to a (3+1)-Dimensional Gross—Pitaevskii Equation with Variable Coefficients

A(3+1)-dimensional Gross-Pitaevskii(GP)equation with time variable coefficients is considered,andis transformed into a standard nonlinear Schr(o|¨)dinger(NLS)equation.Exact solutions of the(3+1)D GP equationare constructed via those of the NLS equation.By applying specific time-modulated nonlinearities,dispersions,andpotentials,the dynamics of the solutions can be controlled.Solitary and periodic wave solutions with snaking andbreathing behavior are reported.

Determination of the geopotential and orthometric height based on frequency shift equation

The orthometric height (OH) system plays a key role in geodesy, and it has broad applications in various fields and activities. Based on general relativity theory (GRT), on an arbitrary equi-geo- potential surface, there does not exist the gravity frequency shift of an electromagnetic wave signal. However, between arbitrary two different equi-geopotential surfaces, there exists the gra- vity frequency shift of the signal. The relationship between the geopotential difference and the gravity frequency shift between arbitrary two points P and Q is referred to as the gravity frequency shift equation. Based on this equation, one can determine the geopotential difference as well as the OH difference between two separated points P and Q either by using electromagnetic wave signals propagated between P and Q, or by using the Global Positioning System (GPS) satellite signals received simultaneously by receivers at P and Q. Suppose an emitter at P emits a signal with frequency f towards a receiver at Q, and the received frequency of the signal at Q is , or suppose an emitter on board a flying GPS satellite emits signals with frequency f towards two receivers at P and Q on ground, and the received frequencies of the signals at P and Q are and , respectively, then, the geopoten-tial dif- ference between these two points can be determined based on the geopotential frequen- cy shift equation, using either the gravity frequency shift ? f or ? , and the corresponding OH difference is further determined based on the Bruns’ formula. Besides, using this approach a unified world height datum system might be realized, because P and Q could be chosen quite arbitrarily, e.g., they are located on two separated continents or islands.

A local refinement purely meshless scheme for time fractional nonlinear Schrodinger equation in irregular geometry region

A local refinement hybrid scheme(LRCSPH-FDM)is proposed to solve the two-dimensional(2D)time fractional nonlinear Schrodinger equation(TF-NLSE)in regularly or irregularly shaped domains,and extends the scheme to predict the quantum mechanical properties governed by the time fractional Gross-Pitaevskii equation(TF-GPE)with the rotating Bose-Einstein condensate.It is the first application of the purely meshless method to the TF-NLSE to the author’s knowledge.The proposed LRCSPH-FDM(which is based on a local refinement corrected SPH method combined with FDM)is derived by using the finite difference scheme(FDM)to discretize the Caputo TF term,followed by using a corrected smoothed particle hydrodynamics(CSPH)scheme continuously without using the kernel derivative to approximate the spatial derivatives.Meanwhile,the local refinement technique is adopted to reduce the numerical error.In numerical simulations,the complex irregular geometry is considered to show the flexibility of the purely meshless particle method and its advantages over the grid-based method.The numerical convergence rate and merits of the proposed LRCSPH-FDM are illustrated by solving several 1D/2D(where 1D stands for one-dimensional)analytical TF-NLSEs in a rectangular region(with regular or irregular particle distribution)or in a region with irregular geometry.The proposed method is then used to predict the complex nonlinear dynamic characters of 2D TF-NLSE/TF-GPE in a complex irregular domain,and the results from the posed method are compared with those from the FDM.All the numerical results show that the present method has a good accuracy and flexible application capacity for the TF-NLSE/GPE in regions of a complex shape.

Spline Solution for the Nonlinear Schrödinger Equation

We develop an exponential spline interpolation method to solve the nonlinear Schr&oumldinger equation. The truncation error and stability analysis of the method are investigated and the method is shown to be unconditionally stable. The conservation quantities are computed to determine the conservation properties of the problem. We will describe the method and present numerical tests by two problems. The numerical simulations results demonstrate the well performance of the proposed method.

一类带阻尼项的Gross-Pitaevskii方程在二维空间中的坍塌性质

研究一类描述吸引Bose Einstein凝聚(BEC)的阻尼Gross Pitaevskii(GP)方程iut=-Δu+｜x｜2u-｜u｜2u+iau,t≥0,x∈R2,a>0.从偏微分方程的严格理论出发,对照BEC的物理性质,应用并发展了M.Tsutsumi(SIAMJMathAnal,1984,15(2):357～366.)中的有关思想和方法,证明了上述GP方程其初值问题将在一定的条件下发生坍塌现象.

具阻尼的Gross-Pitaevskii方程在二维空间中的坍塌性质

讨论出现在吸引玻色-爱因斯坦凝聚中的一类阻尼Gross-Pitaevskii(GP)方程(在数学上又称为带调和势的阻尼非线性Schr dinger方程)itφ+Δφ-|x|2φ+|φ|2φ+iλφ=0,其中t 0,x∈R2,λ是阻尼参数.对照玻色-爱因斯坦凝聚的物理性质,参考R.T.Glassey(J.Math.Phys.,1977,18:1794-1797.)的结果,运用能量方法,得到了一个较为简单的判别条件,当初值满足该条件时,初值问题的解将在有限时间内坍塌.

利用GPS监测电离层不均匀结构探讨

利用上海地区GPS综合应用网提供的高时空分辨率的双频GPS观测资料 ,研究了该区域内一电离层不均匀体的产生、消亡过程 .首先 ,采用Kalman滤波的方法改善双频伪距之差的观测精度 ,并利用参数估计的方法计算该时段内相应的硬件延迟 .再根据电离层单层模型 ,利用GPS双频观测量、测站位置和GPS精密星历 ,求出GPS信号穿刺点的坐标和垂直方向电离层的电子含量 ;然后内插并获取其等值线图 .等值线图随时间的变化表明 ,受等离子体湍流的影响 ,2 0 0 3年 9月 8日北京时间 9时 4 0分左右在 38°N、118°E左右产生了一电离层不均匀体 ,其尺度大约在 5 0km左右 ,生存时间大约为 5min .受地球重力场和高空风场的影响 ,该不均匀体向东北方向扩散 .然后 ,利用大气扩散模型 ,按扩散方程计算分析了该不均匀体可能发生的电离层层区 .理论计算表明 ,该不均匀体发生在电离层扩展F区 ,高度在 35

一种基于卡尔曼滤波的动态目标GPS定位算法

针对动态目标的GPS定位精度不高和实时性较低的问题,通过改进卡尔曼滤波的定位算法,有效消除GPS动态数据的随机误差,给出了仿真得出的运动轨迹对比图和误差曲线对比图.该算法将速度观测量加入到常规的卡尔曼方程中,得出带约束项的改进型卡尔曼方程.通过实验结果可以看出该算法可以有效提高动态目标的GPS定位精度和实时性,具有重要的理论和实际意义.