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可积系统，几何与拓扑。Integrable systems, geometry, and topology.

作为应用，利用屠格式得到了TC方程族的一个新的可积耦合。As the applications, a new integrable coupling of TC hierarchy by using the Tu scheme.

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本文第一章介绍可积系统的哈密顿方法。In chapter one, we give a brief introduction to the Hamiltonian method of integrable system.

在均匀情况下，这是一个可积系统，可以得到该体系的一个严格相图。In free space this is an integrable system, and therefore an exact phase diagram can be obtained.

通过谱问题的双非线性化，导出一个新的可积的辛映射。A new integrable symplectic map is derived by means of nonlinearization of the eigenvalue problem.

证明了函数可积的充要条件是该函数几乎处处连续。It is proved that a function is Riemann integrable if and only if it is continuous almost everywhere.

另外，这一族方程可约化为著名的非线性薛定谔方程。Moreover, it is Liouville integrable and can be reduced to the well-known nonlinear Schrdinger equations.

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特别是系统是不可积系统时，粒子的角分布图为一个不规则的分形。Especially when system is an non integrable one, the distribution of scattered angles is an irregular fractal.

并得到了上述方程可积极小正解的存在性、逼近性、稳定性。Existence, approximation and stability of integrable minimum positive solutions for the above equation are gained.

研究了一族耦合非线性扩散方程，证明了与该扩散方程族对应的每一有限带解都有可积的参数表示。It is proved that any finite band solution to the hierarchy of equations has an integrable parametric representation.

首先给出了C-D对和C-D可积系统的基本理论，然后是具体研究了它们的应用。Firstly, we give a basic notion and basic theory of C-D pair and C-D integrable system and then we study their applications.

研究非共振的拟可积哈密顿系统在受高斯白噪声外激励下的可靠性。In the present paper, the reliability of quasi- integrable -Hamiltonian systems under Gaussian white noises is investigated.

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讨论了非紧对称空间平方可积微分形式的离散序列，并给出其离散谱的一个具体描述。This paper discusses the discrete series of the L 2 integrable differential forms for noncompact Riemannian symmetric spaces.

结果推广了该问题可积的一些原有结果，并给出了通解的参数表示式。Results Original integrable results of this problem are generalized and parametric expression of the general solution is given.

本文举例说明有界可测和L可积函数的巴色伐尔等式不一定成立。This article illustrates that Parseral equality of bounded measurable function and L integrable function is not always realized.

但寻求耦合系统的可积性问题，也仅是由马文秀博士研究的一个孤立子方程的可积耦合，对于一族孤立子方程的可积耦合目前尚未研究。However, the study of integration of coupled system is only the integrable coupling of only one equation due to Doctor Ma Wenxiu.

研究了一个具有同宿轨的二次可积系统在二次保守扰动下的分支现象。The question on bifurcation of limit cycles in quadratic conservative perturbations of a quadratic integrable system is discussed.

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给出两指标局部平方可积强鞅的随机积分的定义，讨论并证明其一些基本性质。In this paper, stochastic integral with two-parameter local square integrable strong martin-sales is defined, its properties are given.

网络的发展使积件库中的资源日益丰富，为教师制作积件式课件提供了丰富的素材资源。The development of internet enriches the resources of integrable ware database, and offers the plentiful material resources for teacher.

采用将圆电流磁矢势表达式中一部分展成级数，使被积函数变成可积分的函数。A part of results on magnetic vector potential of circular loop are expanded, so the integral function is transformed to integrable series.