Multi-Symplectic Splitting Method for Two-Dimensional Nonlinear Schrodinger Equation

Using the idea of splitting numerical methods and the multi-symplectic methods, we propose a multisymplectic splitting （MSS） method to solve the two-dimensional nonlinear Schrodinger equation （2D-NLSE） in this paper. It is further shown that the method constructed in this way preserve the global symplectieity exactly. Numerical experiments for the plane wave solution and singular solution of the 2D-NLSE show the accuracy and effectiveness of the proposed method.

Existence of multi-bump solutions for coupled Schr dinger systems

The Schrodinger equation -△u＋λ2u=｜u｜2q-2u has a unique positive radial solution Uλ, which decays exponentially at infinity. Hence it is reasonable that the Schrolinger system -△u1＋u1=｜u1｜2q-1u1-εb（x）｜u2｜1｜u1｜q-1u1,-△u2＋u2=｜u2｜2q-2u2-εb（x）｜u1｜1｜u2｜q-1u2 has multiple-bump solutions which behave like Uλ in the neighborhood of some points. For u=（u1,u2）∈H1（R3）×H1（R3）, a nonlinear functional Iε（u）=I1（u1）＋I2（u2）-ε/q∫R3b（x）｜u1｜q｜u2｜qdx,is defined,where I1（u1）=1/2｜｜u1｜｜2-1/2q∫R3｜u1｜2qdx and I2（u2）=1/2｜｜u2｜｜2ω-1/2q∫R3｜u2｜2qdx. It is proved that the solutions of the system are the critical points of I,. Let Z be the smooth solution manifold of the unperturbed problem and TzZ is the tangent space. The critical point of I is rewritten as the form of z ＋ w, where w ∈ （TzZ）⊥. Using some properties of Iε, it is proved that there exists a critical point of I, close to the form which is a multi-bump solution.

Study on Application of Hypervirial Therem in the Variational Method

We discuss a methodology problem which is crucially important for solving the Sch?dinger equation in terms of the variational method. We present a complete analysis on the application of the hypervirial theorem for judging the quality of the trial wavefunction without invoking the precise solutions.

Exponential synchronization for delayed nonlinear Schr?dinger equation and applications in optical secure communication

For further exploring the confidentiality of optical communication,exponential synchronization for the delayed nonlinear Schrodinger equation is studied.It is possible for time-delay systems to generate multiple positive Lyapunov exponents without the limitation of system dimension.Firstly,the homoclinic orbit analysis is carried out by using the bifurcation theory,and it is found that there are two homoclinic orbits in the system.According to the corresponding relationship,solitary waves also exist in the system.Secondly,the Melnikov method is used to prove that homoclinic orbits can evolve into chaos under arbitrary perturbations,and then chaotic signals are used as the carriers of information transmission.The Lyapunov exponent spectrum,phase diagram and time series of the system also prove the existence of chaos.Thirdly,an exponential synchronization controller is designed to achieve the chaotic synchronization between the driving system and the response system,and it is proved by the Lyapunov stability theory.Finally,the error system is simulated by using MATLAB,and it is found that the error tends to zero in a very short time.Numerical simulation results demonstrate that the proposed exponential synchronization scheme can effectively guarantee the chaotic synchronization within 1 s.

Wave Functions for Time-Dependent Morse Potentials

The one dimensional Schrodinger equation associated with a time-dependent Morse potentials is studied. We use the invariant operator method （Lewis and Riesenfeld） to obtain approximate solution of the Schrodinger equation in terms of solution of second order ordinary differential equation describes the amplitude of the Morse potentials.

具调和振子的非线性Schrodinger方程

考虑具调和振子的非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程的Ｃａｕｃｈｙ问题，采用Ｇａｌｅｒｋｉｎ方法证 明了整体强解的存在性，使用能量估计方法证明了整体强解的唯一性．

Double-Pole Solution and SolitonAntisoliton Pair Solution of MNLSE/DNLSE Based upon Hirota Method

Hirota method is applied to solve the modified nonlinear Schrodinger equation/the derivative nonlinear Schrodinger equation(MNLSE/DNLSE) under nonvanishing boundary conditions(NVBC) and lead to a single and double-pole soliton solution in an explicit form. The general procedures of Hirota method are presented, as well as the limit approach of constructing a soliton-antisoliton pair of equal amplitude with a particular chirp. The evolution figures of these soliton solutions are displayed and analyzed. The influence of the perturbation term and background oscillation strength upon the DPS is also discussed.

基于Hadamard算子的二维离散量子行走的概率测度估计

二维离散时间量子行走是直线上的量子行走的推广.通过演化算子的作用,行走者能够按照一定规律进行移动.在本文中,我们将Hadamard算子作为控制行走者方向的硬币算子,通过与控制行走者位置的条件转移算子结合,构成完整的演化算子.通过傅里叶变换,将行走者所处的时域空间转换成频域空间后,用傅里叶积分的平稳相位法得到了行走者在t步后处于位置(x,y)的振幅以及此时的概率估计.

Convergence of ground state solutions for nonlinear Schrdinger equations on graphs

We consider the nonlinear Schr¨odinger equation-?u +(λa(x) + 1)u = |u|^(p-1) u on a locally finite graph G =(V, E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ > 1, the equation admits a ground state solution uλ. Moreover, as λ→∞, the solution uλconverges to a solution of the Dirichlet problem-?u + u = |u|^(p-1) u which is defined on the potential well ?. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.

A Novel Approach with Time-Splitting Spectral Technique for the Coupled Schrdinger–Boussinesq Equations Involving Riesz Fractional Derivative

In the present paper the Riesz fractional coupled Schr6dinger-Boussinesq （S-B） equations have been solved by the time-splitting Fourier spectral （TSFS） method. This proposed technique is utilized for discretizing the Schrodinger like equation and further, a pseudospectral discretization has been employed for the Boussinesq-like equation. Apart from that an implicit finite difference approach has also been proposed to compare the results with the solutions obtained from the time-splitting technique. Furthermore, the time-splitting method is proved to be unconditionally stable. The error norms along with the graphical solutions have also been presented here.

A Simple Framework of Conservative Algorithms for the Coupled Nonlinear Schrdinger Equations with Multiply Components

Considering the coupled nonlinear Schr¨odinger system with multiply components, we provide a novel framework for constructing energy-preserving algorithms. In detail, based on the high order compact finite difference method, Fourier pseudospectral method and wavelet collocation method for spatial discretizations, a series of high accurate conservative algorithms are presented. The proposed algorithms can preserve the corresponding discrete charge and energy conservation laws exactly, which would guarantee their numerical stabilities during long time computations.Furthermore, several analogous multi-symplectic algorithms are constructed as comparison. Numerical experiments for the unstable plane waves will show the advantages of the proposed algorithms over long time and verify the theoretical analysis.

A Note on Exact Traveling Wave Solutions of the Perturbed Nonlinear Schrdinger's Equation with Kerr Law Nonlinearity

In this paper,we investigate nonlinear the perturbed nonlinear Schrdinger's equation (NLSE) with Kerr law nonlinearity given in [Z.Y.Zhang,et al.,Appl.Math.Comput.216 (2010) 3064] and obtain exact traveling solutions by using infinite series method (ISM),Cosine-function method (CFM).We show that the solutions by using ISM and CFM are equal.Finally,we obtain abundant exact traveling wave solutions of NLSE by using Jacobi elliptic function expansion method (JEFEM).

Analytical Solutions for the Two-Dimensional Gross-Pitaevskii Equation with a Harmonic Trap

Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation, a nonlinear Schrodinger equation used in describing the system of Bose-Einstein condensates trapped in a harmonic potential. The approximate analytical solutions are obtained successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are agreement very well with each other when the atomic interaction is not too strong.

Analytical Solutions of a Model for Brownian Motion in the Double Well Potential

In this paper, the analytical solutions of Schrodinger equation for Brownian motion in a double well potential are acquired by the homotopy analysis method and the Adomian decomposition method. Double well potential for Brownian motion is always used to obtain the solutions of Fokker-P1anck equation known as the Klein-Kramers equation, which is suitable for separation and additive Hamiltonians. In essence, we could study the random motion of Brownian particles by solving Schr6dinger equation. The anaiytical results obtained from the two different methods agree with each other well The double well potentiai is affected by two parameters, which are analyzed and discussed in details with the aid of graphical illustrations. According to the final results, the shapes of the double well potential have significant influence on the probability density function.

Dynamics of Nonautonomous Dark Solitons

We solve a generalized nonautonomous nonlinear Schrodinger equation analytically by performing the Hirota＇s bilinearization method. The precise expression of a parameter e, which provides a compatibility condition and dark soliton management, is obtained. Comparing with nonautonomous bright soliton, we find that the gain parameter affects both the background and the valley of dark soliton （∈2 ≠ 1） while it has no effects on the wave central position. Moreover, the precise expressions of a nonautonomous black soliton＇s （∈2 = 1） width, background and the trajectory of its wave central, which describe the dynamic behavior of soliton＇s evolution, are investigated analytically. Finally, the stability of the dark soliton solution is demonstrated numerically. It is shown that the main characteristic of the dark solitons keeps unchanged under a slight perturbation in the compatibility condition.