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.     

广义非线性Schrdinger方程的多辛格式与模方守恒律

通过正则变换,构造出广义非线性Schr(o)dinger方程的多辛方程组.对此多辛方程组,导出了一个新的模方守恒多辛格式.数值实验结果表明,多辛格式具有长时间的数值行为,且在保持模方守恒律方面优于蛙跳格式和辛欧拉中点格式.     

广义Zakharov-Kuznetsov方程的对称及精确解

利用改进的直接方法给出了一类广义Zakharov-Kuznetsov方程ut auux bu2ux cuxxx duxyy=0新显式解与旧显式解之间的关系,并且得到了该方程的对称.这些对称推广了已有文献中应用Steinberg s相似方法获得的结果.利用广义Zakharov-Kuznetsov方程新旧显式解之间的关系,本文在已有显式解的基础上给出了方程新的显式解.这些解对于研究某些复杂的物理现象,以及验证数值求解法则的可行性有重要的意义.     

Klein-Gordon-Schrdinger方程的辛Fourier拟谱格式(英文)

主要讨论Klein-Gordon-Schrdinger方程的Fourier拟谱辛格式,包括中点公式和Strmer/Verlet格式.首先构造一个哈密尔顿方程,针对此哈密尔顿方程,在空间方向用Fourier拟谱离散得到一个有限维的哈密尔顿系统,对此有限维系统在时间方向用Strmer/Verlet方法离散得到KGS方程的完全显式的辛格式.中点格式虽然是隐式的但效率也很高,且具有质量守恒律.数值实验表明,辛格式能够在长时间内很好地模拟各类孤立波.     

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.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

孤立波方程的保结构算法

讨论了孤立波方程的保结构差分算法,以一些经典的孤立波方程为例,如KdV,sine-Gordon,K-P方程,给出了它们的辛和多辛结构,说明辛和多辛算法的可适用性．提出局部守恒算法和广义保结构算法的概念,它们是保结构算法的概念自然推广．还给出一种能系统构造局部守恒格式的复合方法．数值例子说明,保结构数值能很好模拟各种孤立波的演化。     

SRLW方程的多辛Fourier谱格式及其守恒律

通过引进正则动量,将对称正则长波方程（简称SRLW方程）转化成多辛形式的方程组,它具有多辛守恒律;介绍了空间方向满足周期边界条件的函数的Fourier谱方法;对SRLW方程的多辛方程组在空间方向利用Fourer谱方法,时间方向上应用Euler中点格式离散,得到其多辛Fourier拟谱格式;证明此格式的一些离散守恒律．用此格式模拟了SRLW方程的单个孤立波,还模拟了多个孤立波的追赶、碰撞和分离过程．     

A Direct Discontinuous Galerkin Method for Nonlinear Schr(o)dinger Equation

讨论一维和二维非线性Schr(o)dinger (NLS)方程的数值求解.基于扩散广义黎曼问题的数值流量,构造一种直接间断Galerkin方法(DDG)求解非线性Schr(o)dinger方程.证明该方法L2稳定性,并说明DDG格式是一种守恒的数值格式.对一维NLS方程的计算表明,DDG格式能够模拟各种孤立子形态,而且可以保持长时间的高精度.二维NLS方程的数值结果显示该方法的高精度和捕捉大梯度的能力.     

非线性Schrdinger方程的直接间断Galerkin方法

讨论一维和二维非线性Schrdinger(NLS)方程的数值求解.基于扩散广义黎曼问题的数值流量,构造一种直接间断Galerkin方法(DDG)求解非线性Schrdinger方程.证明该方法L2稳定性,并说明DDG格式是一种守恒的数值格式.对一维NLS方程的计算表明,DDG格式能够模拟各种孤立子形态,而且可以保持长时间的高精度.二维NLS方程的数值结果显示该方法的高精度和捕捉大梯度的能力.     

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 Method for the Zakharov-Kuznetsov Equation

A new scheme for the Zakharov-Kuznetsov (ZK) equation with the accuracy order of $\mathcal{O}(∆t^2+∆x+∆y^2)$ is proposed. The multi-symplectic conservation property of the new scheme is proved. The backward error analysis of the new multi-symplectic scheme is also implemented. The solitary wave evolution behaviors of the Zakharov-Kunetsov equation are investigated by the new multi-symplectic scheme. The accuracy of the scheme is analyzed.     

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.     

Solutions of Zakharov-Kuznetsov equation with power law nonlinearity in (1+3) dimensions

This paper studies the Zakharov-Kuznetsov equation in (1+3) dimensions with an arbitrary power law nonlinearity. The method of Lie symmetry analysis is used to carry out the integration of the Zakharov-Kuznetsov equation. The solutions obtained are cnoidal waves, periodic solutions, singular periodic solutions, and solitary wave solutions. Subsequently, the extended tanh-function method and the G′/G method are used to integrate the Zakharov-Kuznetsov equation. Finally, the nontopological soliton solution is obtained by the aid of ansatz method. There are numerical simulations throughout the paper to support the analytical development.     

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.     

New Exact Travelling Wave Solutions for Zakharov-Kuznetsov Equation

In this paper, the nonlinear dispersive Zakharov- Kuznetsov equation is solved by using the generalized auxiliary equation method. As a result, new solitary pattern, solitary wave and singular solitary wave solutions are found.     

Compacton and periodic wave solutions of the non-linear dispersive Zakharov-Kuznetsov equation

In this paper, the nonlinear dispersive Zakharov-Kuznetsov equation is solved by using the sine-cosine method. As a result, compactons, periodic, and singular periodic wave solutions are found.      

Deriving the New Traveling Wave Solutions for the Nonlinear Dispersive Equation,KdV-ZK Equation and Complex Coupled KdV System Using Extended Simplest Equation Method

In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov-Kuznetsov (KdV-ZK) equation and complex coupled KdV system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations (NLEEs) in mathematical physics.