Tau Functions of DrinfeldSokolov Hierarchies and Some Relevant Problems 

Author  WuChaoZhong 
Tutor  ZhangYouJin 
School  Tsinghua University 
Course  Mathematics 
Keywords  DrinfeldSokolov hierarchy tau function Hamiltonian structure pseudodifferential operators BKP hierarchy 
CLC  O175.5 
Type  PhD thesis 
Year  2010 
Downloads  21 
Quotes  0 
The DrinfeldSokolov hierarchies, introduced by Drinfeld and Sokolov in 1980’s,are important integrable systems that play a significant role in the development of thesoliton theory as well as in its applications in mathematical physics. In this thesis westudy tau functions of DrinfeldSokolov hierarchies and some relevant problems.Tau functions act as a bridge between integrable systems and related branchesof mathematical physics such as quantum field theory, matrix models, representationtheory and algebraic geometry. In the literature there are several ways to define taufunctions of DrinfeldSokolov hierarchies that admit certain restrictions, but the relation between them are not clear yet. As our first main result, we define tau functions of DrinfeldSokolov hierarchies based on a class of tausymmetric Hamiltoniandensities, and present explicitly the relation between tau functions defined variouslyfor DrinfeldSokolov hierarchies. Such tau functions include those constructed frompseudodi?erential operator representations of the hierarchies, the ones constructed byHollowood and Miramontes for DrinfeldSokolov hierarchies corresponding to a?neLie algebras of ADE type, the ones constructed by Enriquez and Frenkel for hierarchies of mKdV type, as well as those given by Miramontes for generalized DrinfeldSokolov hierarchies and for generating conserved densities of them.Our second main result is that we describe the DrinfeldSokolov hierarchies corresponding to untwisted a?ne Lie algebras of type D and their tau functions in termsof pseudodi?erential operators, then find the bilinear equations satisfied by such taufunctions. These bilinear equations coincide with the integrable hierarchies constructedby Date, Jimbo, Kashiwara, Miwa and by Kac, Wakimoto via the basic representationof untwisted a?ne Lie algebras of type D.Our third main result is the verification of the Dtype simple singularities case ofGivental and Milanov’s conjecture, which was proposed in 2004 to connect singularitytheory and integrable hierarchies. From our result it follows that the GiventalMilanov hierarchies for Dtype simple singularities are equivalent to the DrinfeldSokolov hierarchies corresponding to untwisted a?ne Lie algebras of type D.We also generalize the notion of pseudodi?erential operators, which is crucialto represent the twocomponent BKP hierarchy and its reductions to DrinfeldSokolovhierarchies of type D. Also with the method of Rmatrix, we construct a biHamiltonianstructure of the twocomponent BKP hierarchy, and clarify the relation between theHamiltonian densities and the tau function.