Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

In this paper, we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac (NLD) equation. Based on its multi-symplectic formulation, the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system. Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem, respectively. And the nonlinear subsystem is solved by a symplectic scheme. Finally, a composition method is applied to obtain the final schemes for the NLD equation. We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly. Numerical experiments are presented to show the effectiveness of the proposed methods.     

Multi-Symplectic Wavelet Collocation Method for Maxwell's Equations

In this paper, we develop a multi-symplectic wavelet collocation method for three-dimensional (3-D) Maxwell's equations. For the multi-symplectic formulation of the equations, wavelet collocation method based on autocorrelation functions is applied for spatial discretization and appropriate symplectic scheme is employed for time integration. Theoretical analysis shows that the proposed method is multi-symplectic, unconditionally stable and energy-preserving under periodic boundary conditions. The numerical dispersion relation is investigated. Combined with splitting scheme, an explicit splitting symplectic wavelet collocation method is also constructed. Numerical experiments illustrate that the proposed methods are efficient, have high spatial accuracy and can preserve energy conservation laws exactly.     

Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations

In this paper, we study the integration of Hamiltonian wave equations whose solutions have oscillatory behaviors in time and/or space. We are mainly concerned with the research for multi-symplectic extended Runge–Kutta–Nyström (ERKN) discretizations and the corresponding discrete conservation laws. We first show that the discretizations to the Hamiltonian wave equations using two symplectic ERKN methods in space and time respectively lead to an explicit multi-symplectic integrator (Eleap-frogI). Then we derive another multi-symplectic discretization using a symplectic ERKN method in time and a symplectic partitioned Runge–Kutta method, which is equivalent to the well-known Störmer–Verlet method in space (Eleap-frogII). These two new multi-symplectic schemes are extensions of the leap-frog method. The numerical stability and dispersive properties of the new schemes are analyzed. Numerical experiments with comparisons are presented, where the two new explicit multi-symplectic methods and the leap-frog method are applied to the linear wave equation and the Sine–Gordon equation. The numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation.     

Symplectic Euler Method for Nonlinear High Order Schrödinger Equation with a Trapped Term

In this paper, we establish a family of symplectic integrators for a class of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization. Then we apply the symplectic Euler method to the Hamiltonian system. It is demonstrated that the scheme not only preserves symplectic geometry structure of the original system, but also does not require to resolve coupled nonlinear algebraic equations which is different from the general implicit symplectic schemes. The linear stability of the symplectic Euler scheme and the errors of the numerical solutions are investigated. It shows that the semi-explicit scheme is conditionally stable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical tests suggest that the symplectic integrators are more effective than non-symplectic ones, such as backward Euler integrators.     

Synchronization in nonidentical complex Ginzburg-Landau equations

A cross-correlation coefficient of complex fields has been investigated for diagnosing spatiotemporal synchronization behavior of coupled complex fields. We have also generalized the subsystem synchronization way established in low-dimensional systems to one- and two-dimensional Ginzburg-Landau equations. By applying the indicator to examine the synchronization behavior of coupled Ginzburg-Landau equations, it is shown that our subsystem approach may be of better synchronization performance than the linear feedback method. For the linear feedback Ginzburg-Landau equation, the nonidentical system exhibits generalized synchronization characteristics in both amplitude and phase. However, the nonidentical subsystem may exhibit complete-like synchronization properties. The difference between complex fields for driven and response systems gives a linear scaling with the change of their parameter difference.     

一维Gross-Pitaevskii方程的高阶紧致分裂步多辛格式

首先把一维Gross-Pitaevskli方程改写成多辛Hamiltonian系统的形式,把形式通过分裂变成2个子哈密尔顿系统.然后,对这些子系统用辛或者多辛算法进行离散.通过对子系统数值算法的不同组合方式,得到不同精度的具有多辛算法特征数值格式.这些格式不仅具有多辛格式、分裂步方法和高阶紧致格式的特征,而且是质量守恒的.数值实验验证了新格式的数值行为.     

An improved symplectic precise integration method for analysis of the rotating rigid–flexible coupled system

This paper presents an improved symplectic precise integration method (PIM) to increase the accuracy and keep the stability of the computation of the rotating rigid–flexible coupled system. Firstly, the generalized Hamilton's principle is used to establish a coupled model for the rotating system, which is discretized and transferred into Hamiltonian systems subsequently. Secondly, a suitable symplectic geometric algorithm is proposed to keep the computational stability of the rotating rigid–flexible coupled system. Thirdly, the idea of PIM is introduced into the symplectic geometric algorithm to establish a symplectic PIM, which combines the advantages of the accuracy of the PIM and the stability of the symplectic geometric algorithm. In some sense, the results obtained by this method are analytical solutions in computer for a long span of time, so the time-step can be enlarged to speed up the computation. Finally, three numerical examples show the stability of computation, the accuracy of solving stiff equations and the capability of solving nonlinear equations, respectively. All these examples prove the symplectic PIM is a promising method for the rotating rigid–flexible coupled systems.     

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.     

Symplectic wavelet collocation method for Hamiltonian wave equations

This paper introduces a novel symplectic wavelet collocation method for solving nonlinear Hamiltonian wave equations. Based on the autocorrelation functions of Daubechies compactly supported scaling functions, collocation method is conducted for the spatial discretization, which leads to a finite-dimensional Hamiltonian system. Then, appropriate symplectic scheme is employed for the integration of the Hamiltonian system. Under the hypothesis of periodicity, the properties of the resulted space differentiation matrix are analyzed in detail. Conservation of energy and momentum is also investigated. Various numerical experiments show the effectiveness of the proposed method.     

广义Zakharov-Kuznetsov方程的多辛Preissmann格式

广义Zakharov-Kuznetsov 方程作为一类重要的非线性方程有着许广泛的应 用前景,基于Hamilton 空间体系的多辛理论研究了广义Zakharov-Kuznetsov方程的数值 解法,讨论了利用Preissmann 方法构造离散多辛格式的途径, 并构造了一种典型的半隐 式的多辛格式, 该格式满足多辛守恒律、局部能量守恒律. 数值算例结果表明该多辛离 散格式具有较好的长时间数值稳定性.     

广义Zakharov-Kuznetsov方程的多辛Preissmann格式

广义Zakharov-Kuznetsov方程作为一类重要的非线性方程有着许多广泛的应用前景,基于Hamilton空间体系的多辛理论研究了广义Zakharov-Kuznetsov方程的数值解法,讨论了利用Preissmann方法构造离散多辛格式的途径,并构造了一种典型的半隐式的多辛格式,该格式满足多辛守恒律、局部能量守恒律.数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性.     

Strongly coupled intracavity frequency-doubled pulsed lasers

Simple expressions for the peak power, pulse energy, and pulse length of strongly coupled intracavity frequency-doubled lasers are presented. For nonlinear coupling values that correspond to many common systems, these expressions agree well with the results of numerical integration of the rate equations. In contrast with weak nonlinear or linear coupling, pulsed lasers with strong nonlinear coupling only deplete the population inversion down to the threshold level.     

Multi-symplectic method for the coupled Schrodinger-KdV equations

In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schrtidinger-KdV equations （CS＇KE） with periodic boundary conditions. Then we develop a novel multi-symplectic Fourier pseudospectral （MSFP） scheme for the CSKE. In numerical experiments, we compare the MSFP method with the Crank-Nicholson （CN） method. Our results show high accuracy, effectiveness, and good ability of conserving the invariants of the MSFP method.     

Explicit multi-symplectic methods for Klein–Gordon–Schrödinger equations

In this paper, we propose explicit multi-symplectic schemes for Klein–Gordon–Schrödinger equation by concatenating suitable symplectic Runge–Kutta-type methods and symplectic Runge–Kutta–Nyström-type methods for discretizing every partial derivative in each sub-equation. It is further shown that methods constructed in this way are multi-symplectic and preserve exactly the discrete charge conservation law provided appropriate boundary conditions. In the aim of the commonly practical applications, a novel 2-order one-parameter family of explicit multi-symplectic schemes through such concatenation is constructed, and the numerous numerical experiments and comparisons are presented to show the efficiency and some advantages of the our newly derived methods. Furthermore, some high-order explicit multi-symplectic schemes of such category are given as well, good performances and efficiencies and some significant advantages for preserving the important invariants are investigated by means of numerical experiments.     

基于同伦技术的Burgers方程的小波精细积分算法

以Burgers方程为例,结合区间小波精细积分方法,将同伦摄动方法的应用范围推广到多维非线性问题,给出一种求解非线性偏微分方程的新的小波精细积分方法,得到一种近似解析解的数值结果,对时间步长不敏感,更适合于求解非线性问题.     

A Multi-Symplectic Compact Method for the Two-Component Camassa-Holm Equation with Singular Solutions

The two-component Camassa-Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa-Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.     

Multi-symplectic Birkhoffian structure for PDEs with dissipation terms

A generalization of the multi-symplectic form for Hamiltonian systems to self-adjoint systems with dissipation terms is studied. These systems can be expressed as multi-symplectic Birkhoffian equations, which leads to a natural definition of Birkhoffian multi-symplectic structure. The concept of Birkhoffian multi-symplectic integrators for Birkhoffian PDEs is investigated. The Birkhoffian multi-symplectic structure is constructed by the continuous variational principle, and the Birkhoffian multi-symplectic integrator by the discrete variational principle. As an example, two Birkhoffian multi-symplectic integrators for the equation describing a linear damped string are given.     

Free vibrations of an uncertain energy pumping system

The aim of this paper is to study the energy pumping (the irreversible energy transfer from one structure, linear, to another structure, nonlinear) robustness considering the uncertainties of the parameters of a two DOF mass-spring-damper, composed of two subsystems, coupled by a linear spring: one linear subsystem, the primary structure, and one nonlinear subsystem, the so-called NES (nonlinear energy sink). Three parameters of the system will be considered as uncertain: the nonlinear stiffness and the two dampers. Random variables are associated to the uncertain parameters and probability density functions are constructed for the random variables applying the Maximum Entropy Principle. A sensitivity analysis is then performed, considering different levels of dispersion, and conclusions are obtained about the influence of the uncertain parameters in the robustness of the system.     

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

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

Conformal structure-preserving method for damped nonlinear Schrdinger equation

In this paper,we propose a conformal momentum-preserving method to solve a damped nonlinear Schrodinger(DNLS) equation.Based on its damped multi-symplectic formulation,the DNLS system can be split into a Hamiltonian part and a dissipative part.For the Hamiltonian part,the average vector field(AVF) method and implicit midpoint method are employed in spatial and temporal discretizations,respectively.For the dissipative part,we can solve it exactly.The proposed method conserves the conformal momentum conservation law in any local time-space region.With periodic boundary conditions,this method also preserves the total conformal momentum and the dissipation rate of momentum exactly.Numerical experiments are presented to demonstrate the conservative properties of the proposed method.