“Good” Boussinesq方程的多辛算法

考虑非线性“Good”Boussinesq方程的多辛形式,对于多辛形式,提出了一个新的等价于中心Preissman积分的15点多辛格式。数值试验结果表明：多辛格式具有良好的长时间数值行为。     

Nonlocal Symmetries and Explicit Solutions of the Boussinesq Equation

The nonlocal symmetry of the Boussinesq equation is obtained from the known Lax pair. The explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetry. Some other types of solutions, such as rational solutions and error function solutions, are given by using the fourth Painlev′e equation with special values of the parameters. For some interesting solutions, the figures are given out to show their properties.     

Symplectic Structure of Discrete Hamiltonian Systems

This paper is concerned with the symplectic structure of discrete nonlinear Hamiltonian systems. The results are related to an open problem that was first proposed by C. D. Ahlbrandt [J. Math. Anal. Appl.180 (1993), 498-517] discussed elsewhere in the literature. But we give a different statement and different proof. Under a solvable condition, we show that the solution operator of a discrete nonlinear Halmiltonian system is symplectic. Then its phase flow is a discrete one-parameter family of symplectic transformations and preserves the phase volume.     

Group-Invariant Solutions and Conservation Laws of One-Dimensional Nonlinear Wave Equation

Based on classical Lie symmetry method, the one-dimensional nonlinear wave equation is investigated. By using four-dimensional subalgebras of the equation, the invariant groups and commutator table are constructed. Furthermore, optimal system of the equation is obtained, and the exact solutions can be gained by solving reduced equations. Finally, a complete derivation of the conservation law is given by using conservation multipliers.     

A New Viscous Regularization of the Riemann Problem for Burgers' Equation

This paper gives a new viscous regularization of the Riemann problem for Burgers' equation u_t + (\frac{u²}{2})_z = 0 with Riemann initial data u = u_(x ≤ 0), u = u_+(x > 0} at t = 0. The regularization is given by u_t + (\frac{u²}{2})_z = εe^tu_{zz} with appropriate initial data. The method is different from the classical method, through comparison of three viscous equations of it. Here it is also shown that the difference of the three regularizations approaches zero in appropriate integral norms depending on the data as ε → 0_+ for any given T > 0.     

扩充偏微分方程(组)守恒律和对称的辅助方程方法及微分形式吴方法的应用

首先,我们给出了引入伴随方程(组)扩充原方程(组)的策略使给定偏微分方程(组)的扩充方程组具有对应泛瓯即,成为Lagrange系统的方法,以此为基础提出了作为偏微分方程(组)传统守恒律和对称概念的一种推广-偏微分方程(组)扩充守恒律和扩充对称的概念;其次,以得到的Lagrange系统为基础给定了确定原方程(组)扩充守恒律和扩充对称的方法,从而达到扩充给定偏微分方程(组)的首恒律和对称的目的;第三,提出了适用于一般形式微分方程(组)的计算固有守恒律的方法;第四,实现以上算法过程中,我们先把计算(扩充)守恒律和对称问题均归结为求解超定线性齐次偏微分方程组(确定方程组)的问题.然后,对此关键问题我们提出了用微分形式吴方法处理的有效算法;最后,作为方法的应用我们计算确定了非线性电报方程组在内的五个发展方程(组)的新守恒律和对称,同时也说明了方法的有效性.     

New exact solutions of a generalised Boussinesq equation with damping term and a system of variant Boussinesq equations via double reduction theory

The conservation laws of a generalised Boussinesq (GB) equation with damping term are derived via the partial Noether approach. The derived conserved vectors are adjusted to satisfy the divergence condition. We use the definition of the association of symmetries of partial differential equations with conservation laws and the relationship between symmetries and conservation laws to find a double reduction of the equation. As a result, several new exact solutions are obtained. A similar analysis is performed for a system of variant Boussinesq (VB) equations.     

Variational Poisson-Nijenhuis structures for partial differential equations

We explore variational Poisson-Nijenhuis structures on nonlinear partial differential equations and establish relations between the Schouten and Nijenhuis brackets on the initial equation and the Lie bracket of symmetries on its natural extensions (coverings). This approach allows constructing a framework for the theory of nonlocal structures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 268–282, February, 2008.     

Bretherton方程的对称约化和精确解

运用李群对称方法解决Bretherton方程问题,得到方程的对称约化和群不变解,比如幂级数解,最后得出该问题的守恒率.     

扰动Boussinesq方程的近似守恒律

构造了具有扰动项的Boussinesq方程的近似守恒向量和近似守恒律.在方程允许拉格朗日函数的情况下,利用欧拉方程的部分拉格朗日函数方法,研究了含有一阶线性组合扰动项的Boussineq方程的近似守恒律.给出了该方程的近似守恒向量及近似守恒律的分类结果.     

The Prolongation Structures and Nonlocal Symmetries for Modified Boussinesq System

The Wahlquist-Estabrook (WE) prolongation structures of modified Boussinesq (MB) system are studied from the coverings point of view. The realizations and classifications of one-dimensional coverings of this system are obtained completely. Moreover the sufficient and necessary conditions for a vector field to be a nonlocal symmetry of this system are also demonstrated in the WE prolongation structures.     

弹性平面扇形域问题及哈密顿体系*

通过变量代换及变分原理,将平面弹性扇形域的方程导向哈密顿体系,从而可用分离变量法、本征函数展开等方法求解扇形域的分析单元,这样便可以与有限元的程序系统相结合。显示了哈密顿体系、辛数学的应用潜力。     

A molecular dynamics model for symplectic intergrators

Mechanical systems in the very large scale like in celestial mechanics or in the very small scale like in the molecular dynamics can be modelled without dissipation. The resulting Hamiltonian systems possess conservation properties, which are characterized with the term symplecticness, Numerical integration schemes should preserve the symplecticness. Different methods are introduced and their performance is studied for constant and variable step size. As test examples two systems from molecular dynamics are introduced.     

Decomposing a new nonlinear differential-difference system under a Bargmann implicit symmetry constraint

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.     

一维新的Hamilton可积方程

构造了Loop代数~A_{-1}的一个子代数,利用屠格式导出了一族新的可积孤子方程族,并且是Liouville可积系,具有双Hamilton结构。     

Energy Conservation with Non-Symplectic Methods: Examples and Counter-Examples

Energy conservation of numerical integrators is well understood for symplectic one-step methods. This article provides new insight into energy conservation with non-symplectic methods. Sufficient conditions and counter-examples are presented. AMS subject classification (2000) 65L06, 65P10, 37J99.Submitted June 2004. Accepted October 2004. Communicated by Syvert Nørsett.     

Solvability of a Boundary Value Problem for a Differential Equation of the Boussinesq Type

We study the solvability and the construction of the solution of a boundary value problem with a nonlocal integral boundary condition for a three-dimensional analog of the fourth-order homogeneous Boussinesq type differential equation. Separation of variables is used to derive a criterion for the unique solvability of this nonlocal problem. The problem is also considered in the case of violation of the unique solvability criterion.     

Lie symmetries and conservation laws of the Fokker-Planck equation with power diffusion

We concentrate on Lie symmetries and conservation laws of the Fokker-Planck equation with power diffusion describing the growth of cell populations. First, we perform a complete symmetry classification of the equation, and then we find some interesting similarity solutions by means of the symmetries and the variable coefficient heat equation. Local dynamical behaviors are analyzed via the solutions for the growing cell populations. Second, we show that the conservation law multipliers of the equation take the form Λ=Λ(t,x,u), which satisfy a linear partial differential equation, and then give the general formula of conservation laws. Finally, symmetry properties of the conservation law are investigated and used to construct conservation laws of the reduced equations.     

哈密尔顿系统的有限元法

利用常微分方程的连续有限元法,结合函数的M-型展开,对非线性哈密尔顿系统证明了连续一、二次有限元分在3阶量、5阶量意义下近似保辛,且保持能量守恒.在数值实验中结合庞加莱截面,哈密尔顿混沌数值试验结果与理论相吻合.