（2+1）维耗散长波方程与(2+1)维Broer-Kaup方程新的类多孤子解

进一步拓广齐次平衡法的应用，并对关键的操作步骤进行了改进，从而简便地求出了（2+1 ）维耗散长波方程和（2+1）维Broer-Kaup方程新的类多孤子解-这种解更具有一般性，它包 含着已有文献给出的类多孤子解- 关键词： 齐次平衡法 类多孤子解 （2+1）维耗散长波方程 （2+1）维Broer-Kaup方程     

(2+1)维色散长波方程的新的类孤子解

应用一种新的修改的代数方法去求解(2+1)维色散长波方程,获得方程的大量新的精确解.这些解包括类孤子解、类周期解、类有理解、类双曲函数解、类Jacobi椭圆函数解等等. 关键词： (2+1)维色散长波方程 类孤子解 类有理解 类双曲函数解 类Jacobi椭圆 函数解     

长水波近似方程的多孤子解

利用直接而简单的齐次平衡方法，给出了长水波近似方程的多孤子解.本方法可进一步推广研究一大类非线性波动方程. 关键词：     

构造非线性发展方程的无穷序列复合型类孤子新解

为了构造非线性发展方程的无穷序列复合型类孤子新解, 进一步研究了G'(ξ)/G(ξ) 展开法. 首先, 给出一种函数变换, 把常系数二阶齐次线性常微分方程的求解问题转化为一元二次方程和Riccati方程的求解问题. 然后, 利用Riccati方程解的非线性叠加公式, 获得了常系数二阶齐次线性常微分方程的无穷序列复合型新解. 在此基础上, 借助符号计算系统Mathematica, 构造了改进的(2+1)维色散水波系统和(2+1)维色散长波方程的无穷序列复合型类孤子新精确解. 关键词： G'(ξ)/G(ξ)展开法')" href="#">G'(ξ)/G(ξ)展开法 非线性叠加公式 非线性发展方程 复合型类孤子新解     

非线性孤子方程的齐次平衡法

齐次平衡法是求非线性孤子方程孤波解的一种十分有效的方法.对齐次平衡法的一些关键步骤进行拓广使用，获得了非线性孤子方程一批新的具有更为丰富形式的精确解，孤波解仅是其中的一种特殊情形，使得结果更加完美. 关键词：     

Gerdjikov-Ivanov方程的精确解

研究在量子场理论、弱非线性色散水波、非线性光学等领域中出现的Gerdjikov-Ivanov方程.对Gerdjikov-Ivanov方程的研究会导出具有高次非线性项的非线性数学物理方程.选取Liénard方程作为辅助常微分方程,借助于它并根据齐次平衡原则,求解了Gerdjikov-Ivanov方程,得到了该方程的包络孤立波解和包络正弦波解. 关键词： 齐次平衡原则 F展开法 Gerdjikov-Ivanov方程 包络孤立波解     

高阶色散效应常系数Ginzburg-Landau方程自相似脉冲演化的解析分析

采用自相似分析方法,基于常系数高阶色散的Ginzburg-Landau方程,通过分离变量法得出了高阶色散效应自相似脉冲演化的解析解,给出了自相似脉冲的振幅、相位、啁啾以及脉冲宽度的一般表达式.研究表明,在增益光纤的二阶正常色散区域,同时考虑高阶色散和增益色散双重效应影响下演化的自相似孤子脉冲仍然保持线性啁啾;振幅解析解的三阶色散效应显著.这与数值计算的结果非常一致. 关键词： 三阶色散 Ginzburg-Landau方程 自相似脉冲 二阶正常色散     

两个非线性发展方程的双向孤波解与孤子解

采用分步确定拟解的原则, 对齐次平衡法求非线性发展方程孤子解的关键步骤作了进一步改 进. 以广义Boussinesq方程和bidirectional Kaup-Kupershmidt方程为应用实例, 说明使用 该方法可有效避免“中间表达式膨胀”的问题, 除获得标准Hirota形式的孤子解外, 还能获 得其他形式的孤子解. 关键词： 齐次平衡法 孤子解 孤波解 广义Boussinesq方程 bidirectional Kaup-Kupershmi dt方程     

势形式破裂孤子方程的dromion孤子解结构

使用改进的齐次平衡方法,研究了破裂孤子方程的孤子解结构,发现它具有单孤子解,单曲线孤子解,单dromion孤子解,多dromion孤子解。     

构造非线性发展方程无穷序列类孤子精确解的一种方法

辅助方程法已构造了非线性发展方程的有限多个新精确解. 本文为了构造非线性发展方程的无穷序列类孤子精确解, 分析总结了辅助方程法的构造性和机械化性特点. 在此基础上,给出了一种辅助方程的新解与Riccati方程之间的拟Bäcklund变换. 选择了非线性发展方程的两种形式解,借助符号计算系统 Mathematica,用改进的(2+1) 维色散水波系统为应用实例,构造了该方程的无穷序列类孤子新精确解. 这些解包括无穷序列光滑类孤子解, 紧孤立子解和尖峰类孤立子解.     

New exact periodic solutions to （2＋1）-dimensional dispersive long wave equations

In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.     

New multi—soliton solutions and travelling wave solutions of the dispersive long—wave equations

Using the extended homogeneous balance method,the (1 1)-dimensional dispersive long-wave equations have been solved.Starting from the homogeneous balance method,we have obtained a nonlinear transformation for simplifying a dispersive long-wave equation into a linear partial differential equation.Usually,we can obtain only a type of soliton-like solution.In this paper,we have further found some new multi-soliton solutions and exact travelling solutions of the dispersive long-wave equations from the linear partial equation.     

一类非线性色散方程中的新型奇异孤立波

研究一类非线性色散广义DGH方程的新型奇异孤立波及其Painlevé可积性.利用Painlevé分析发现当对流项强度m=2时广义DGH方程是可积的,这是一个新的可积方程.通过构造新的变量代换以及auto-Backlund变换获得该方程丰富的奇异孤立波解,如紧孤立波（compacton）、尖峰孤立波（peakon）、新型带尖点的双孤立波和带爆破点的双孤立波等. 关键词： 非线性色散方程 可积性 奇异孤立波     

New solitons and periodic wave solutions for the dispersive long wave equations

We make use of Jacobi elliptic functions to construct periodic wave solutions for the dispersive long wave equations. In the limit cases the multiple soliton solutions are also obtained. The properties of some periodic and soliton solutions for the dispersive long wave equations are shown by some figures. As a result, we can successfully obtain the solitary wave solutions that can be found by the previous work.     

Normal forms for weakly dispersive wave equations

A normal form is derived for a unidirectional solution to weakly dispersive nonlinear wave equations, and based on this form, we discuss the nonintegrability of the original equations.     

Coupled scalar field equations for nonlinear wave modulations in dispersive media

A review of the generic features as well as the exact analytical solutions of a class of coupled scalar field equations governing nonlinear wave modulations in dispersive media like plasmas is presented. The equations are derivable from a Hamiltonian function which, in most cases, has the unusual property that the associated kinetic energy is not positive definite. To start with, a simplified derivation of the nonlinear Schrödinger equation for the coupling of an amplitude modulated high-frequency wave to a suitable low-frequency wave is discussed. Coupled sets of time-evolution equations like the Zakharov system, the Schrödinger-Boussinesq system and the Schrödinger-Korteweg-de Vries system are then introduced. For stationary propagation of the coupled waves, the latter two systems yield a generic system of a pair of coupled, ordinary differential equations with many free parameters. Different classes of exact analytical solutions of the generic system of equations are then reviewed. A comparison between the various sets of governing equations as well as between their exact analytical solutions is presented. Parameter regimes for the existence of different types of localized solutions are also discussed. The generic system of equations has a Hamiltonian structure, and is closely related to the well-known Hénon-Heiles system which has been extensively studied in the field of nonlinear dynamics. In fact, the associated generic Hamiltonian is identically the same as the generalized Hénon-Heiles Hamiltonian for the case of coupled waves in a magnetized plasma with negative group dispersion. When the group dispersion is positive, there exists a novel Hamiltonian which is structurally same as the generalized Hénon-Heiles Hamiltonian but with indefinite kinetic energy. The above correspondence between the two systems has been exploited to obtain the parameter regimes for the complete integrability of the coupled waves. There exists a direct one-to-one correspondence between the known integrable cases of the generic Hamiltonian and the stationary Hamiltonian flows associated with the only integrable nonlinear evolution equations (of polynomial and autonomous type) with a scale-weight of seven. The relevance of the generic system to other equations like the self-dual Yang-Mills equations, the complex Korteweg-de Vries equation and the complexified classical dynamical equations has also been discussed.     

A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation

In this paper, we present an approach for seeking exact solutions with coefficient function forms of conformable fractional partial differential equations. By a combination of an under-determined fractional transformation and the Jacobi elliptic equation, exact solutions with coefficient function forms can be obtained for fractional partial differential equations. The innovation point of the present approach lies in two aspects. One is the fractional transformation, which involve the traveling wave transformations used by many articles as special cases. The other is that more general exact solutions with coefficient function forms can be found, and traveling wave solutions with constants coefficients are only special cases of our results. As of applications, we apply this method to the space-time fractional (2+1)-dimensional dispersive long wave equations and the time fractional Bogoyavlenskii equations. As a result, some exact solutions with coefficient function forms for the two equations are successfully found.     

耗散旋波介质中的色散关系

应用Maxwell经典电磁理论，结合旋波介质的本构关系，研究耗散旋波介质中光波的色散关系．结果表明，色散关系与介质电导率、旋波因子等因素密切相关．     

光子晶体光纤超连续谱产生过程中色散波的孤子俘获研究

基于光子晶体光纤中脉冲演化遵循的非线性薛定谔方程, 用数值模拟的方法分别研究了飞秒脉冲在单零色散点和双零色散点光子晶体光纤中超连续谱的产生和色散波的孤子俘获现象. 结果表明: 与单零色散点光子晶体光纤相比, 双零色散点光子晶体光纤产生的超连续谱既包含了蓝移色散波, 又包含了红移色散波, 且当满足群速度匹配时, 孤子通过四波混频不仅能俘获蓝移色散波, 而且能俘获红移色散波, 从而产生新的俘获波频谱成分. 为了清楚地观察脉冲传输的时频特性, 通过模拟交叉相关频率分辨光学开关技术, 得到了孤子俘获色散波的演化过程. 关键词： 超连续谱 色散波 孤子俘获 光子晶体光纤     

Smoothing properties and retarded estimates for some dispersive evolution equations

Smoothing properties, in the form of space-time integrability properties, play an important role in the study of dispersive evolution equations. A number of them follow from a combination of general arguments and specific estimates. We present a general formulation which makes the separation between the two types of ingredients as clear as possible, and we illustrate it with the examples of the Schrödinger equation, of the wave equation, and of a class of 1+1 dimensional equations related to the Benjamin-Ono equation. Of special interest for the Cauchy problem are retarded estimates expressed in terms of those properties. We derive a number of such estimates associated with the last example, and we mention briefly an application of those estimates to the Cauchy problem for the generalized Benjamin-Ono equation.Laboratoire associé au Centre National de la Recherche Scientifique