非线性耦合标量场方程的新双周期解(Ⅱ)

基于具有双周期解的常微分方程，提出了一种构造非线性微分方程双周期解的新方法，在计算机符号软件帮助下方法可实现机械化.应用此方法于非线性耦合标量场方程，得到了该方程的大量的新精确解. 关键词： 非线性耦合标量场方程 双周期解 精确解     

微扰的耦合非线性薛定谔方程的近似求解

将直接微扰方法应用于可积的含修正项的非线性薛定谔方程，通过近似解与精确解的比较确定了直接微扰方法的可靠性.继而，将该方法应用于微扰的耦合非线性薛定谔方程，并获得了该微扰方程的可靠的近似解. 关键词： 直接微扰方法 微扰 耦合非线性薛定谔方程 近似解     

非线性耦合标量场方程的精确解

在非线性耦合标量场方程已有精确解基础上，利用适当的函数变换方法，再次获得几种精确解，从而新旧结果一起构成耦合标量场方程的8种精确解，其中有6种孤子解，另外两种为三角函数形式的周期解.讨论了这些结果在物理学其他几个著名方程上的应用. 关键词：     

非线性耦合标量场方程的新双周期解(Ⅰ)

对于具有丰富物理意义和众多应用价值的非线性耦合标量场方程，利用sinh-Gordon方程展开法，导出了五种新的双周期解，同时在退化的情形得到了新的孤波解和三角函数解.结果说明该方程具有丰富的解的结构，方程描述的物理模型所具有的物理现象仍将是物理学家和数学家进一步探讨的对象. 关键词： 非线性耦合标量场方程 sinh-Gordon方程展开法 双周期解 精确解     

利用耦合的Riccati方程组构造微分-差分方程精确解

通过引入耦合的Riccati方程组得到一个构造非线性微分-差分方程精确解的代数方法.作为实例,将该方法应用到了一般格子方程,相对论的Toda格子方程和(2+1)维Toda格子方程.借助符号计算软件Mathematica,获得了这些方程的扭结型孤波解和复数解.该方法也适合求解其他非线性微分-差分方程的精确解. 关键词： 耦合Riccati方程组 格子方程 相对论的Toda格子方程 (2+1)维Toda格子方程     

构造非线性发展方程精确解的一种方法

在双曲正切函数法、齐次平衡法、辅助方程法的基础上引入非线性发展方程的一个新形式解和新辅助方程，并利用符号计算系统Mathematica构造了Benjamin-Bona-Mahoney(BBM)方程和修正的 BBM方程的新精确孤立波解.这种方法在寻找其他非线性发展方程的新精确解方面具有普遍意义. 关键词： 新辅助方程 形式解 非线性发展方程 精确孤立波解     

光纤中变系数非线性Schr(o)dinger方程的孤子解及其应用

基于推广的立方非线性Klein—Gordon方程对一般形式的变系数非线性Schrodinger方程进行研究，讨论了无啁啾情形的孤子解，发现了包括亮、暗孤子解和类孤子解在内的一些新的精确解．同时对基本孤子的色散控制方法进行了简单讨论．作为特例，常系数非线性Schrodinger方程和两类特殊的变系数非线性Schrodinger方程的结果和已知的形式一致．此外，还研究了一个周期增益或损耗的光纤系统，得到了有意义的结果．     

修正的BBM方程新的精确解

应用进一步修正的简单方程法对修正的 Benjamin -Bona -Mahoney (mBBM )方程进行求解,给出了mBBM方程新的精确类孤波解,取定某些参数值,便可得到精确孤波解.这种方法也可用于寻找其它常系数以及变系数非线性发展方程(组)的精确解,具有一定的普适性.     

非线性Schroedinger方程组的精确解组

运用Maple语言程序，在没有假设的条件下，得到了具有耦合特性的非线性Schrfidinger方程组的行波精确解组及其约束条件方程，它们的表达式涵盖了所有的耦合解组与非耦合解组，具有任意性。耦合解组的算例函数及其特性分析，解释了α螺旋蛋白质螺旋链运动模型的行波孤立子解的耦合效应，揭示了增加、稳定和控制蛋白质活性和功能的方向。文章的研究方法，为求解耦合的非线性微分方程组的行波精确解组探索了蹊径。     

修正的BBM方程新的精确解北大核心CSCD

应用进一步修正的简单方程法对修正的Benjamin-Bona-Mahoney(mBBM)方程进行求解,给出了mBBM方程新的精确类孤波解,取定某些参数值,便可得到精确孤波解.这种方法也可用于寻找其它常系数以及变系数非线性发展方程(组)的精确解,具有一定的普适性.     

关于分离变量法的一个注记

从B-cklund变换出发，利用Cole-Hopf变换，一些有重要物理意义的(2+1)维非线性演化方程(组)被简化为具有两个分别关于(x,t),(y,t)的任意函数的单个线性偏微分方程.通过对该方程解的研究，获得了原方程一些新的精确解.其中，一些近年来被广泛研究的由分离变量法所获得的解也被重新得到. 关键词： B-cklund变换 Cole-Hopf变换 分离变量法 精确解     

Solving Integrable Broer-Kaup Equations in （2＋1）-Dimensional Spaces via an Improved Variable Separation Approach

Starting from Baecklund transformation and using Cole-Hopf transformation, we reduce the integrable Broer-Kaup equations in (2 1)-dimensional spaces to a simple linear evolution equation with two arbitrary functions of {x, t} and {y, t} in this paper. And we can obtain some new solutions of the original equations by investigating the simple nonlinear evolution equation, which include the solutions obtained by the variable separation approach.     

Solving Integrable Broer-Kaup Equations in (2+1)-Dimensional Spaces via an Improved Variable Separation Approach

Starting from Backlund transformation and using Cole-Hopf transformation, we reduce the integrable Broer-Kaup equations in (2 1)-dimensional spaces to a simple linear evolution equation with two arbitrary functions of {x,t} and {y,t} in this paper. And we can obtain some new solutions of the original equations by investigating the simple nonlinear evolution equation, which include the solutions obtained by the variable separation approach.     

A Hopf-like equation and perturbation theory for differential delay equations

We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion.     

同伦分析法在求解非线性演化方程中的应用

利用同伦分析法求解了(2+1)维改进的 Zakharov-Kuznetsov方程, 得到了它的近似周期解,该解与精确解符合很好. 结果表明，同伦分析法在求解高维非线性演化方程时, 仍然是一种行之有效的方法. 同时,还对该方法进行了一定的扩展， 经过扩展后的方法能够更方便地求解更多非线性演化方程的高精度近似解析解. 关键词： 同伦分析法 改进的 Zakharov-Kuznetsov方程 周期解     

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.     

Geometric Quantization and Two Dimensional QCD

In this article, we will discuss geometric quantization of two dimensional Quantum Chromodynamics with fermionic or bosonic matter fields. We identify the respective large-N _c phase spaces as the infinite dimensional Grassmannian and the infinite dimensional Disc. The Hamiltonians are quadratic functions, and the resulting equations of motion for these classical systems are nonlinear. In [33], it was shown that the linearization of the equations of motion for the Grassmannian gave the 't Hooft equation. We will see that the linearization in the bosonic case leads to the scalar analog of the 't Hooft equation found in [36]. Received: 13 August 1996 / Accepted: 20 May 1997     

Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

In this paper, a new methodology is formulated for solving the reduced Fokker‐Planck (FP) equations in high dimensions based on the idea that the state space of large‐scale nonlinear stochastic dynamic system is split into two subspaces. The FP equation relevant to the nonlinear stochastic dynamic system is then integrated over one of the subspaces. The FP equation for the joint probability density function of the state variables in another subspace is formulated with some techniques. Therefore, the FP equation in high‐dimensional state space is reduced to some FP equations in low‐dimensional state spaces, which are solvable with exponential polynomial closure method. Numerical results are presented and compared with the results from Monte Carlo simulation and those from equivalent linearization to show the effectiveness of the presented solution procedure. It attempts to provide an analytical tool for the probabilistic solutions of the nonlinear stochastic dynamics systems arising from statistical mechanics and other areas of science and engineering.     

A symmetry-preserving difference scheme for high dimensional nonlinear evolution equations

In this paper, a procedure for constructing discrete models of the high dimensional nonlinear evolution equanons is presented. In order to construct the difference model, with the aid of the potential system of the original equation and compatibility condition, the difference equations which preserve all Lie point symmetries can be obtained. As an example, invariant difference models of the （2＋1）-dimensional Burgers equation are presented.     

Kinetic equation for a dense soliton gas

We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons due to soliton-soliton collisions. Owing to complete integrability of the soliton equations, only pairwise soliton interactions contribute to the solution, and the evolution reduces to a transport of the eigenvalues of the associated spectral problem with the corresponding soliton velocities modified by the collisions. The proposed general procedure of the derivation of the kinetic equation is illustrated by the examples of the Korteweg-de Vries and nonlinear Schr?dinger (NLS) equations. As a simple physical example, we construct an explicit solution for the case of interaction of two cold NLS soliton gases.