摘要
本文主要介绍孤立子方程的可积系统(即非线性演化方程族的生成及可积性质和非线性演化方程族的扩展可积模型)和非线性方程的精确求解。第一章概述了孤立子理论的产生和发展、研究概况及其研究意义。在第二章中，首先利用外积的性质构造了一个3M维的loop代数(?)_M，由此可设计出许多新的等谱问题，作为应用，本文得到了一个多分量可积的Boite-Pempinelli-Tu(BPT)族。其次，运用2+1维的零曲率方程和屠格式得到了一类2+1维的多分量的可积系。作为一个实例，得到了一类广义2+1维Kaup-Newell(KN)族。最后利用3维的Lie代数构造出相应的loop代数，由此建立一个广义的等谱问题，运用屠格式直接获得了多分量的KN方程族，作为约化情形分别得到了广义Burgers方程和广义耦合kdv方程。作为可积系统的进一步研究是可积耦合问题，即非线性演化方程族的一类扩展可积模型。在第三章中，首先将第二章中的loop代数扩展为新的高维的loop代数，由此设计恰当的等谱问题，利用屠格式求出了第二章方程族(2．3．7)中所得的可积系的相应的扩展可积模型。然后以已有的一个Lie代数的子代数为基础，通过线性组合得到了一个5维的Lie代数，然后构造出相应的loop代数，由此建立一个广义的等谱问题，运用屠格式和零曲率方程获得了第二章第四节中KN方程族的扩展可积系统，给出了求可积耦合的一种简便方法，这种方法可以普遍使用。在第四章中，首先介绍了非线性方程求解的各种方法，然后重点介绍了齐次平衡原则，最后，作为应用，给出了Fisher方程的精确行波解。
This paper introduce with emphisis the integrable system of the solitonequations include: the formulation and integrability of the nonlinear evolutionhierarchy as well as their expanding integrable systems and the exact solutionsof the nonlinear equations. In the first chapter, historical origin and someresearches of soliton theory together with its research meaning are presented.In the second chapter, firstly, a loop algebra (?)_M with 3M dimensions isconstructed by using of some properties of exterior algebra, which is devotedto establishing many isospectral problems. As its application, a multi-componentsystem BPT hierarchy is obtained. Secondly, a type of (2+1)-dimensionalmulti-component integrable hierarchy is obtained with the help of a(2+1)-dimensional zero-curvature equation and Tu scheme. As an applicationexample, we obtain a generalized (2+1)-dimensional KN hierarchy; at last In termsof the Lie algebra with 3-dimensional, we establishanew loop algebra (?)_M, thenthe extended integrable model of the muti-component KN hierarchy is presentedby use of the generalized isospectral problem and Tu sheme which reduces to ageneralized Burgers equation and generalized coupled kdv equation. In the thirdchapter, firstly, new expanding loop algebras of the loop algebras presented inthe equation systems (2.3.16) of second chapter are constructed, then acorresponding expanding integrable models are engendered by employing properisospectral problems and Tu sheme. Secondly, a 5-dimensional Lie algebra isestablished with the help of the linear combination of a subalgebra, in termsof the Lie algebra, we establish a new loop algebra, then the extended integrablemodel of the muti-component hierarchy (2.4.7) is presented by use of thegeneralized isospectral problem and Tu sheme. The simple approach to generateexpanding integrable models can be used generally. In the fourth chapter, firstintroduced simply the methods of the exact solutions of nonlinear equations, thenintroduced with empnasis the nomogeneous balance principle, finally, thkes the application, has given the exact traveling solutions of the Fisher equation.
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