表象变换和久期微扰理论在耦合杜芬方程中的应用

本文通过对耦合杜芬方程线性项的表象变换及非线性项的久期微扰理论的应用,将耦合杜芬方程转化为简正表象下的退耦合形式,由此可以很方便地得出耦合杜芬方程的解.为了验证该方法的正确性,设计了音叉耦合实验,观测到了振幅谱谱峰的劈裂以及"振滞回线"现象,这些实验结果都可以和之前所得的理论结果符合得很好.本文求解耦合非线性方程的方法,为灵活运用非线性理论提供一种方案,同时可以推广到光、电等耦合体系,对理解耦合体系的动力学行为具有一定的指导意义.     

自由汇流旋涡多相耦合输运演变机理

含自由液面的汇流旋涡抽吸演变中存在多相耦合、物质传输、能量剧烈交换等物理过程,其中所涉及的多相流体耦合输运机理是具有高度非线性特征的复杂动力学问题,多相黏滞耦合输运动力学建模与数值求解具有较高难度.针对上述问题,提出一种含自由液面的汇流旋涡多相耦合输运建模与求解方法.基于水平集-流体体积耦合(CLSVOF)计算方法,结合连续表面张力模型和可实现(k-ε)湍流模型,建立含自由液面的汇流旋涡多相耦合输运动力学模型;利用一种有效的体积修正方案来计算高速旋转多相流,保证流场质量守恒和无散度的速度场;结合相间耦合求解策略对多相流体分布与多相界面进行精确追踪.基于旋流场多特征物理变量,得到多相耦合界面动态演变与跨尺度涡团输运规律,揭示了多相耦合输运过程与压力脉动特性之间的相互作用机理.研究结果表明:多相耦合输运过程是流体介质过渡的关键状态,旋涡微团受到不同时空扰动模式在界面处形成层层螺纹波形;旋涡多相耦合输运过程随着水口尺度增大而增强,且耦合能量激波引起非线性压力脉动现象.研究结果可为旋涡输运机理、涡团跨尺度求解、流型追踪等方面的研究提供有益借鉴.     

地-气耦合动力系统的近似解析解

本文研究了一类地-气耦合系统的非线性模式.利用同伦映射方法，将问题转化为线性问题求解.最后得到了原非线性问题的近似表示式. 关键词： 非线性 地-气耦合 动力系统 同伦映射     

磁场与流场耦合问题中Newton-Laphson方法的收敛性分析(英文)

针对磁场与流场耦合问题的数值分析,提出并证明求解离散化过程所得到的非线性方程组牛顿-拉夫逊方法的-类局部收敛性条件.这-条件不仅给出了时间步长与空间步长、拟压缩因子等之间的关系,而且为数值求解磁场与流场耦合问题的牛顿-拉夫逊方法收敛性提供了理论依据.数值算例表明时间步长的实际取值要比理论值偏大.     

可积系统的双线性约化方法

本综述主要介绍了双线性约化方法在可积系统求解中的应用.这一方法基于双线性方法和解的双Wronskian表示.对于通过耦合系统约化而获得的可积方程,先求解未约化的耦合系统,给出用双Wronskian表示的解;进而利用双Wronskian的规则结构,施以适当的约化技巧,获得约化后的可积方程的解.以非线性Schr?dinger方程族和微分-差分非线性Schrodinger方程为具体例证,详述此方法的应用技巧.除了经典可积方程,该方法也适用于非局部可积系统的求解.其他例子还包括Fokas-Lenells方程和非零背景的非线性Schr?dinger方程等可积系统的求解.     

一类高维耦合的非线性演化方程的简单求解

利用一个简单的变换，一类高维耦合的非线性演化方程可以被约化为一低维的简单方程，将已有的求解法应用于简单方程，十分简捷的获得了原方程大量的精确解. 关键词： 非线性耦合方程 精确解 tanh函数方法     

隐-显积分因子间断Galerkin方法求解二维辐射扩散方程

提出一种求解二维非平衡辐射扩散方程的数值方法.空间离散上采用加权间断Galerkin有限元方法,其中数值流量的构造采用一种新的加权平均;时间离散上采用隐-显积分因子方法,将扩散系数线性化,然后用积分因子方法求解间断Galerkin方法离散后的非线性常微分方程组.数值试验中在非结构网格上求解了多介质的辐射扩散方程.结果表明:对于强非线性和强耦合的非线性扩散方程组,该方法是一种非常有效的数值算法.     

用参数微扰法求解非线性谐振子问题

用参数微扰法求解了一维非线性谐振子及二维非线性耦合谐振子的一级近似本征能量.计算表明,参数微扰法的基态和激发态结果比相应的无参数微扰法的一级近似甚至多级近似结果精度有显著提高.     

耗散Jaynes-Cummings模型中微观与唯象学主方程方法的比较

在有耗散的Jaynes-Cummings模型中,对原子与腔复合系统初始处于贝尔态或是原子激发态而腔场为真空态时分别运用微观与唯象学主方程进行求解,并分析这两类主方程在不同耦合区域的区别．结果显示,在强耦合和大失谐情况下,微观主方程求解得到的原子基态的布居数呈线性增长,但唯象学主方程获得的结果却呈非线性增加．在弱耦合区域,这两类主方程求解得到的原子基态布居数都出现快速增加,但是用唯象学主方程要比用微观主方程得到的布居数增加得更快．通过分析与讨论,该结果可能对耗散的腔QED中更好的实现逻辑门提供理论参考．     

耗散Jaynes-Cummings模型中微观与唯象学主方程方法的比较

在有耗散的Jaynes-Cummings模型中,对原子与腔复合系统初始处于贝尔态或是原子激发态而腔场为真空态时分别运用微观与唯象学主方程进行求解,并分析这两类主方程在不同耦合区域的区别.结果显示,在强耦合和大失谐情况下,微观主方程求解得到的原子基态的布居数呈线性增长,但唯象学主方程获得的结果却呈非线性增加.在弱耦合区域,这两类主方程求解得到的原子基态布居数都出现快速增加,但是用唯象学主方程要比用微观主方程得到的布居数增加得更快.通过分析与讨论,该结果可能对耗散的腔QED中更好的实现逻辑门提供理论参考.     

Exact Solutions of a Coupled Burgers System

The exact solutions of a new coupled Burgers system are studied in three different ways. The first type of solutions are found thanks to the coupled Burgers system possessing a simple single Burgers reduction. The second type of multiple soliton solutions are revealed via the decouple procedure. The third type of exact solutions are found by means of a prior ansatz and solutions of the heat conduction equation. Two different kinds of soliton fission phenomena of the model are discovered and a special type of completely elastic soliton collision without phase shift of the model is also displayed.     

Special solutions for nonlinear wave-particle interaction

A class of exact solutions for a coupled set of nonlinear equations describing the interaction between two propagating waves and a system of particles is found. These solutions include traveling wave solutions of the non-linear coupled equations.     

The auxiliary elliptic-like equation and the exp-function method

The auxiliary equation method is very useful for finding the exact solutions of the nonlinear evolution equations. In this paper, a new idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the auxiliary elliptic-like equation are derived using exp-function method, and then the exact solutions of the nonlinear evolution equations are derived with the aid of auxiliary elliptic-like equation. As examples, the RKL models, the high-order nonlinear Schrödinger equation, the Hamilton amplitude equation, the generalized Hirota-Satsuma coupled KdV system and the generalized ZK-BBM equation are investigated and the exact solutions are presented using this method.     

非线性波方程求解的新方法

从Legendre椭圆积分和Jacobi椭圆函数的定义出发，得到了新的变换，并把它用于非线性演化方程的求解.用三个具体的例子，如非线性Klein-Gordon方程、Boussinesq方程和耦合的mKdV方程组，说明了具体的求解步骤.比较方便地得到非线性演化方程或方程组的新解析解，如周期解、孤子解等. 关键词： Jacobi椭圆函数 非线性方程 周期解 孤子解     

New Exact Solutions and Special Coherent Structures for Coupled Integrable Dispersionless Equation

Using improved homogeneous balance method, we obtain new exact solutions for the coupled integrable dispersionless equation. On the basis of these exact solutions, we find some new interesting coherent structures by selecting arbitrary functions appropriately.     

Rotating 5D-Kaluza-Klein Space-Times from Invariant Transformations

Using invariant transformations of the five-dimensional Kaluza-Klein (KK) field equations, we find a series of formulae to derive axially symmetric stationary exact solutions of the KK theory starting from static ones. The procedure presented in this work allows to derive new exact solutions up to very simple integrations. Among other results, we find exact rotating solutions containing magnetic monopoles, dipoles, quadrupoles, etc., coupled to scalar and to gravitational multipole fields.     

Exact periodic wave solutions to the coupled Schrödinger-KdV equation and DS equations

The exact solutions for the coupled non-linear partial differential equations are studied by means of the mapping method proposed recently by the author. Taking the coupled Schrödinger-KdV equation and DS equations as examples, abundant periodic wave solutions in terms of Jacobi elliptic functions are obtained. Under the limit conditions, soliton wave solutions are given.     

Class of Einstein–Maxwell phantom fields: rotating and magnetized wormholes

Using a new ansatz for solving the Einstein equations with a scalar field with inverted sign in the kinetic term (phantom field), I find a series of formulae to derive axial symmetric stationary exact solutions of the phantom fields in general relativity. I focus on the solutions which represent wormholes. The procedure presented in this work allows to derive new exact solutions up to very simple integrations. Among other results, I find exact rotating solutions containing magnetic monopoles, dipoles, etc., coupled to phantom scalar and to gravitational multipole fields.     

非线性耦合微分方程组的精确解析解

提出了利用耦合的Riccati方程组的某些特解构造非线性微分方程组精确解析解的一种方法.应用这种方法研究了两个耦合的常微分方程组,系统地获得了它们的一些精确解.给出了非线性浅水波近似方程组和非线性Schr?dinger-KdV方程组若干新的孤波解. 关键词： 非线性耦合方程组 Riccati方程组 符号计算 孤波     

Exact traveling wave solutions for a generalized Hirota-Satsuma coupled KdV equation by Fan sub-equation method

In this Letter, the Fan sub-equation method is used to construct exact solutions of a generalized Hirota-Satsuma coupled KdV equation. Many exact traveling wave solutions are successfully obtained, which contain more general solitary wave solutions and Jacobian elliptic function solutions with double periods. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.