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     

一般变换下的Jacobi椭圆函数展开法及应用

将在行波变换下的Jacobi椭圆函数展开法推广到范围非常广泛的一般函数变换下进行，利用这一方法求得了一些非线性发展方程的精确周期解，这些解包括了在行波变换下所求得的周期解. 证明了一些非线性发展方程的周期解一定是行波解. 关键词： 非线性发展方程 周期解 行波解 Jacobi椭圆函数     

一般变换下双Jacobi椭圆函数展开法及应用

将行波变换下修正的双Jacobi椭圆函数展开法推广到范围广泛的一般函数变换下进行.利用这一方法求得了一类非线性方程更多新的周期解，这些解包括了在行波变换下所求得的周期解. 关键词： Jacobi椭圆函数展开法 非线性发展方程 函数变换 周期解     

(n+1)维双Sine-Gordon方程的新精确解

给出包含第一种椭圆方程的三角函数型辅助方程及其解的叠加公式.在一般函数变换下,借助符号计算系统Mathematica,构造了(n+1)维双sine-Gordon方程新的Jacobi椭圆函数精确解.这些解包括了行波变换下的Jacobi椭圆函数精确解、精确孤立波解和三角函数解.     

变系数KP方程新的类孤波解和解析解

用普通Sine-Gordon的行波变换方程，提出了一种新的求解变系数Kaolomtsev-Petviashvili(KP)方程的方法，获得了变系数KP方程新的类孤波解、类Jacobi椭圆函数解和三角函数解. 关键词： 变系数KP方程 Sine-Gordon方程 类椭圆函数解 类孤波解     

非线性发展方程的丰富的Jacobi椭圆函数解

通过把十二个Jacobi椭圆函数分类成四组，提出了新的广泛的Jacobi椭圆函数展开法，利用这一方法求得了非线性发展方程的丰富的Jacobi椭圆函数双周期解.当模数m→0或1时，这些解退化为相应的三角函数解或孤立波解和冲击波解. 关键词： 非线性发展方程 Jacobi椭圆函数 双周期解 行波解     

立方非线性Schr?dinger方程的Weierstrass椭圆函数周期解

利用Weierstrass椭圆函数展开法对非线性光学、等离子体物理等许多系统中出现的立方非线性 Schr(o)dinger方程进行了研究.首先通过行波变换将方程化为一个常微分方程,再利用Weierstrass椭圆函数展开法思想将其化为一组超定代数方程组,通过解超定方程组,求得了含Weierstrass椭圆函数的周期解,以及对应的Jacobi椭圆函数解和极限情况下退化的孤波解.该方法有以下两个特点:一是可以借助数学软件Mathematica自动地完成;二是可以用于求解其它的非线性演化方程(方程组).     

变系数广义KdV方程新的类孤波解和精确解

用普通KdV方程作变换，构造变系数广义KdV方程的解，获得了变系数广义KdV方程新的Jacobi椭圆函数精确解、类孤波解、三角函数解和Weierstrass椭圆函数解. 关键词： KdV方程 变系数广义KdV方程 类孤波解 精确解     

立方非线性Schr(o)dinger方程的Weierstrass椭圆函数周期解

利用Weierstrass椭圆函数展开法对非线性光学、等离子体物理等许多系统中出现的立方非线性Schr(o)dinger方程进行了研究.首先通过行波变换将方程化为一个常微分方程,再利用Weierstrass椭圆函数展开法思想将其化为一组超定代数方程组,通过解超定方程组,求得了含Weierstrass椭圆函数的周期解,以及对应的Jacobi椭圆函数解和极限情况下退化的孤波解.该方法有以下两个特点:一是可以借助数学软件Mathematica自动地完成;二是可以用于求解其它的非线性演化方程(方程组).     

立方非线性Schrdinger方程的Weierstrass椭圆函数周期解

利用Weierstrass椭圆函数展开法对非线性光学、等离子体物理等许多系统中出现的立方非线性Schrdinger方程进行了研究.首先通过行波变换将方程化为一个常微分方程,再利用Weierstrass椭圆函数展开法思想将其化为一组超定代数方程组,通过解超定方程组,求得了含Weierstrass椭圆函数的周期解,以及对应的Jacobi椭圆函数解和极限情况下退化的孤波解.该方法有以下两个特点:一是可以借助数学软件Mathematica自动地完成;二是可以用于求解其它的非线性演化方程(方程组).     

A theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations. III The Markovian-non-Markovian mix

The proof of a theorem that allows one to construct deterministic evolution equations from a set, with two subsets, containing two types of discrete stochastic evolution equation is developed. One subset evolves Markovianly and the other non-Markovianly. As an illustrative example, the deterministic evolution equations of quantum electrodynamics are derived from two sets of Markovian and non-Markovian stochastic evolution equations, of different type, after an average over realization, using the theorem. This example shows that deterministic differential equations that contain both first-order and second-order time derivatives can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations. Two explicit examples, the first containing updating rules that depend on one previous time step and the second containing updating rules that depend on two previous time steps, are given in detail in order to show step by step the linear transformations that allow one to obtain the deterministic differential equations.     

Some linear Bäcklund transformations

Dirac's equation is a first-order auto-Bäcklund transformation for the Klein-Gordon equation in 4 variables, this equation being second-order in each of the 4 variables. Here first-order auto-Bäcklund transformations are given for any linear equation of arbitrary (possibly different) order in each of n variables (n arbitrary). This class of equations includes, for example, any of the linearised versions of the KdV hierarchy or equations, and the n-dimensional diffusion equation.     

Localized waves in three-component coupled nonlinear Schrdinger equation

We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.     

Majorana Transformation for Differential Equations

We present a method for reducing the order of ordinary differential equations satisfying a given scaling relation (Majorana scale-invariant equations). We also develop a variant of this method, aimed to reduce the degree of nonlinearity of the lower order equation. Some applications of these methods are carried out and, in particular, we show that second-order Emden–Fowler equations can be transformed into first-order Abel equations. The work presented here is a generalization of a method used by Majorana in order to solve the Thomas–Fermi equation.     

关于变摆长的单摆运动与相关的铅直运动的耦合

研究了一个变摆长单摆运动的例子,在平面极坐标系下给出了描述该运动的二阶微分方程组.采用微分方程高精度数值解法得到了随时间变化的摆角、摆长、摆长变化率以及运动轨迹的数值精确解.在小摆角近似下,采用迭代方法推导出摆角、摆角变化率的一级近似解析表达式和摆长、摆长变化率的二级近似解析表达式.由解析表达式得到的数值结果与数值精确解相比较,二者在前几个摆动周期内相吻合.     

FOLDING TRANSFORMATIONS AND HKY MAPPINGS

We present a new method for the derivation of mappings of HKY type. These are second-order mappings which do not have a biquadratic invariant like the QRT mappings, but rather an invariant of degree higher than two in at least one of the variables. Our method is based on folding transformations which exist for some discrete Painlevé equations. They are transformations which relate the variable of a discrete Painlevé equation to the square of the variable of some other one. By considering the autonomous limit of these relations we derive folding-like transformations which relate QRT mappings to HKY ones. We construct the invariants of the latter mappings and show how they can be extended beyond the ones given by the strict application of the folding transformation.     

Higher-Order Rogue Wave Pairs in the Coupled Cubic-Quintic Nonlinear Schrödinger Equations

We study some novel patterns of rogue wave in the coupled cubic-quintic nonlinear Schrödinger equations. Utilizing the generalized Darboux transformation, the higher-order rogue wave pairs of the coupled system are generated. Especially, the first-and second-order rogue wave pairs are discussed in detail. It demonstrates that two classical fundamental rogue waves can be emerged from the first-order case and four or six classical fundamental rogue waves from the second-order case. In the second-order rogue wave solution, the distribution structures can be in triangle, quadrilateral and ring shapes by fixing appropriate values of the free parameters. In contrast to single-component systems, there are always more abundant rogue wave structures in multi-component ones. It is shown that the two higher-order nonlinear coefficients ρ₁ and ρ₂ make some skews of the rogue waves.     

线性常微分方程的保线性变换及其应用

研究了线性常微分方程的保线性变换，得到任意两个二阶线性常微分方程等价的条件，并用于求解一类二阶线性变系数齐次常微分方程．对数学物理方法教学中怎样通过适当的变换把给定的二阶线性变系数齐次常微分方程化为可解的方程给出了合理解释。     

A SYSTEMATIC METHOD OF FINDING LINEARIZING TRANSFORMATIONS FOR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS II: EXTENSION TO COUPLED ODEs

In this second paper on the method of deriving linearizing transformations for nonlinear ODEs, we extend the method to a set of two coupled second-order nonlinear ODEs. We show that beside the conventional point, Sundman and generalized linearizing transformations one can also find a large class of mixed or hybrid type linearizing transformations like point-Sundman, point-generalized linearizing transformation and Sundman-generalized linearizing transformation in coupled second-order ODEs using the integrals of motion. We propose suitable algorithms to identify all these transformations (with maximal in number) in a straightforward manner. We illustrate the method of deriving each one of the linearizing transformations with a suitable example.     

INVARIANT LINEARIZATION CRITERIA FOR SYSTEMS OF CUBICALLY NONLINEAR SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Invariant linearization criteria for square systems of second-order quadratically nonlinear ordinary differential equations (ODEs) that can be represented as geodesic equations are extended to square systems of ODEs cubically nonlinear in the first derivatives. It is shown that there are two branches for the linearization problem via point transformations for an arbitrary system of second-order ODEs and its reduction to the simplest system. One is when the system is at most cubic in the first derivatives. One obtains the equivalent of the Lie conditions for such systems. We explicitly solve this branch of the linearization problem by point transformations in the case of a square system of two second-order ODEs. Necessary and sufficient conditions for linearization to the simplest system by means of point transformations are given in terms of coefficient functions of the system of two second-order ODEs cubically nonlinear in the first derivatives. A consequence of our geometric approach of projection is a rederivation of Lie's linearization conditions for a single second-order ODE and sheds light on more recent results for them. In particular we show here how one can construct point transformations for reduction to the simplest linear equation by going to the higher space and just utilizing the coefficients of the original ODE. We also obtain invariant criteria for the reduction of a linear square system to the simplest system. Moreover these results contain the quadratic case as a special case. Examples are given to illustrate our results.