Drinfeld-Sokolov 方程簇是 Drinfeld 和 Sokolov 在上世纪八十年代初提出的一类重要的可积系统, 它们对孤子理论的发展及其在数学物理中的应用有着极为重要的意义. 本文研究的是 Drinfeld-Sokolov 方程簇的 tau 函数及某些相关问题.Tau 函数是联系可积系统与量子场论、矩阵模型、表示论、代数几何等数学物理分支中相关问题的重要桥梁. Drinfeld-Sokolov 方程簇的 tau 函数已有多种定义方法, 它们各有限制条件, 且相互关系有待明确. 本文的第一个主要结果是通过构造 Drinfeld-Sokolov 方程簇满足 tau 对称条件的 Hamilton 密度及由此定义的 tau 函数, 本文给出了定义不同的 tau 函数之间的明确关系, 其中包括利用方程簇的拟微分算子表示来定义的 tau 函数, Hollowood 和 Miramontes 对 A-D-E 型仿射 Lie 代数对应的 Drinfeld-Sokolov 方程簇定义的 tau 函数, Enriquez 和 Frenkel 定义的 mKdV 类方程簇的 tau 函数, 以及 Miramontes 对广义 Drinfeld-Sokolov 方程簇构造的可生成其守恒密度的 tau 函数.本文的第二个主要结果是利用拟微分算子给出了非扭 D 型仿射 Lie 代数对应的 Drinfeld-Sokolov 方程簇及其 tau 函数的表示, 并给出了 tau 函数所满足的双线性方程. 这些双线性方程正是 Date, Jimbo, Kashiwara, Miwa 以及 Kac, Wakimoto 通过非扭 D 型仿射 Lie 代数的基本表示来构造的可积方程簇.Givental 和 Milanov 在 2004 年提出一个联系奇点理论与可积方程簇的猜想, 证明这个猜想对应 D 型单奇点的情形是本文的第三个主要结果. 由此我们得到 D 型单奇点对应的 Givental-Milanov 方程簇等价于非扭 D 型仿射 Lie 代数对应的 Drinfeld-Sokolov 方程簇.本文的还拓广了拟微分算子的定义. 我们用这些算子来表示二分量 BKP 方程簇及其到 D 型 Drinfeld-Sokolov 方程簇的约化, 再借助 R-矩阵方法构造了二分量 BKP 方程簇的双 Hamilton 结构, 并给出其 Hamilton 密度与 tau 函数的关系.

The Drinfeld-Sokolov hierarchies, introduced by Drinfeld and Sokolov in 1980's, are important integrable systems that play a significant role in the development of the soliton theory as well as in its applications in mathematical physics. In this thesis we study tau functions of Drinfeld-Sokolov hierarchies and some relevant problems.Tau functions act as a bridge between integrable systems and related branches of mathematical physics such as quantum field theory, matrix models, representation theory and algebraic geometry. In the literature there are several ways to define tau functions of Drinfeld-Sokolov hierarchies that admit certain restrictions, and the relation between them are not clear yet. As our first main result, we define tau functions of Drinfeld-Sokolov hierarchies based on a class of tau-symmetric Hamiltonian densities, and present explicitly the relation between tau functions defined variously for Drinfeld-Sokolov hierarchies. Such tau functions include those constructed from pseudo-differential operator representations of the hierarchies, the ones constructed by Hollowood and Miramontes for Drinfeld-Sokolov hierarchies corresponding to affine Lie algebras of A-D-E type, the ones constructed by Enriquez and Frenkel for hierarchies of mKdV type, as well as the ones given by Miramontes for generalized Drinfeld-Sokolov hierarchies and for generating conserved densities of them.Our second main result is that we describe the Drinfeld-Sokolov hierarchies corresponding to untwisted affine Lie algebras of type D and their tau functions in terms of pseudo-differential operators, hence find the bilinear equations satisfied by such tau functions. These bilinear equations coincide with the integrable hierarchies constructed by Date, Jimbo, Kashiwara, Miwa as well as Kac, Wakimoto via the basic representation of untwisted affine Lie algebras of type D.Our third main result is the verification of the D-type simple singularities case of Givental and Milanov's conjecture, which was proposed in 2004 to connect singularity theory and integrable hierarchies. From our result it follows that the Givental-Milanov hierarchies for D-type simple singularities are equivalent to the Drinfeld-Sokolov hierarchies corresponding to untwisted affine Lie algebras of type D.We also generalize the notion of pseudo-differential operators, which is crucial to represent the two-component BKP hierarchy and its reductions to Drinfeld-Sokolov hierarchies of type D. Also with the method of R-matrix, we construct a bi-Hamiltonian structure of the two-component BKP hierarchy, and clarify the relation between the Hamiltonian densities and the tau function.