The reduction problem, Automorphic Lie Algebra and Integrable systems
报告人:
Prof. Alexander V. Mikhailov
Department of Applied Mathematics , University of Leeds, UK
报告时间:
2010-11-04 16:00
报告地点:
理科楼1304报告厅
主办单位:
数学科学系
简介:
We study a new class of in_nite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic Lie algebras. It is also a natural generalisation of the construction used by Victor Kac in his study of graded Lie algebras. In contrast to the Kac-Moody algebras, automorphic Lie algebras are quasi-graded and, in a certain sense, are deformations of the Kac-Moody Lie algebras. We discuss the progress in the classi_cation problem for automorphic Lie algebras corresponding to _nite groups of automorphisms (_nite reduction groups). Integrable systems related to quasi-graded Lie algebras are nonhomogeneous and are deformations of well known integrable systems in the graded case. The two dimensional generalisation of the Volterra chain is an interesting example of such a system. Its continuous limit is the famous Kadomtsev-Petviashvili equation. Using the dressing method we study exact solutions of the 2-d Volterra system. Apart of soliton-like solutions we have found exact solutions corresponding to wave fronts. Classi_cation of soliton and wave front solutions is related to the Schubert decomposition of a Grassmanian.
