Multi-symplectic variational integrators for nonlinear Schrdinger equations with variable coefficients

In this paper, we propose a variational integrator for nonlinear Schrdinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrdinger equations with variable coefficients, cubic nonlinear Schrdinger equations and Gross–Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.     

多模海森堡代数的算符恒等式

利用算符代数中的分析方法,得到了多模海森堡(Heisenberg)代数中的BCH公式和压缩算符的展开式。     

A high order energy preserving scheme for the strongly coupled nonlinear Schrōdinger system

A high order energy preserving scheme for a strongly coupled nonlinear Schrōdinger system is roposed by using the average vector field method. The high order energy preserving scheme is applied to simulate the soliton evolution of the strongly coupled Schrōdinger system. Numerical results show that the high order energy preserving scheme can well simulate the soliton evolution, moreover, it preserves the discrete energy of the strongly coupled nonlinear Schrōdinger system exactly.     

The backward phase flow and FBI-transform-based Eulerian Gaussian beams for the Schrödinger equation

We propose the backward phase flow method to implement the Fourier–Bros–Iagolnitzer (FBI)-transform-based Eulerian Gaussian beam method for solving the Schrödinger equation in the semi-classical regime. The idea of Eulerian Gaussian beams has been first proposed in [12]. In this paper we aim at two crucial computational issues of the Eulerian Gaussian beam method: how to carry out long-time beam propagation and how to compute beam ingredients rapidly in phase space. By virtue of the FBI transform, we address the first issue by introducing the reinitialization strategy into the Eulerian Gaussian beam framework. Essentially we reinitialize beam propagation by applying the FBI transform to wavefields at intermediate time steps when the beams become too wide. To address the second issue, inspired by the original phase flow method, we propose the backward phase flow method which allows us to compute beam ingredients rapidly. Numerical examples demonstrate the efficiency and accuracy of the proposed algorithms.     

“Eigen-Operator” Method of the Square of Schroedinger Operator in Interaction Picture and Its Application in Describing Parametric Amplifiers

We extend the method of searching “eigen-operator” of the square of the Schroedinger operator to the interaction picture, which not only helps to construct Hamiltonians of two kinds of parametric amplifiers but also leads to a new uncertainty relation regarding to the free Hamiltonian and the interacting Hamiltonian.     

Multi-linear Variable Separation Approach to Solve a (2+1)-Dimensional Generalization of Nonlinear Schrödinger System

By using a Bäcklund transformation and the multi-linear variable separation approach, we find a new general solution of a (2+1)-dimensional generalization of the nonlinear Schrödinger system. The new “universal” formula is defined, and then, rich coherent structures can be found by selecting corresponding functions appropriately.     

Nonlinear Squeezed States and Multiplication Rule of Nonlinear Squeezing Operators Gained via Nonlinear Coherent State Representation and IWOP Technique

Using the nonlinear coherent state representation we derive nonlinear squeezed states and the multiplication rule of nonlinear squeezing operators. We find that the symplectic matrices multiplication rule in nonlinear coherent state projection operator representation maps into the multiplication rule of successive nonlinear squeezing operators.The technique of integral within an ordered product of operators plays an essential role in deriving the multiplication rule.     

一般阻尼随机微分方程的拟局部振荡算法

提出了一种模拟随机微分方程的拟局部振荡算法,即利用算符劈裂方法和势函数的泰勒展开,对噪声作用下耗散粒子的时间演化算符进行分解,得到了对应涨落行为的扩散算符和对应确定轨迹的漂移算符.其中局部简谐势场的涨落过程可获得解析解,而剩余的确定项则利用简单的Euler算法积分.应用到几个算例并与常用的两种算法相比较,结果表明:本算法随时间步长最稳定,可使用较大的时间步长.     

Electron localization of H_2~+ in a dc electric field

A dc electric field is utilized to steer the electron motion after the molecular ion H_2~+ is excited by an ultrashort ultraviolet laser pulse. The numerical simulation shows that the electron localization distribution and the dissociation control ratio are dependent on the polarization direction and amplitude of the dc electric field. Most electrons of the dissociation state move opposite to the dc electric field and stabilize at the dressed-up potential well, for the dressed-down well is occupied by the electrons of the 1 sσ_g state.