2004 Fiscal Year Final Research Report Summary
Soliton equations and combinatories
Project/Area Number 
15540165

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  KYOTO UNIVERSITY 
Principal Investigator 
SHIOTA Takahiro Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20243008)

Project Period (FY) 
2003 – 2004

Keywords  KP hierarchy / BKP hierarchy / CalogeroMoser system / tau function / matrix model 
Research Abstract 
As is wellknown, the motion of poles of a KP rational solution obeys the hierarchy of CalogeroMoser dynamical systems (and similarly for the trigonometric and elliptic solutions of KP hierarchy). This fact may get renewed interest after recent discovery by J.F.Van Diejen on the relation between the zeros of KdV wave function and the RuijsenaarsSchneider system. Both the KP and the CalogeroMoser hierarchies allow generalizations with various internal symmetries from Lie theory point of view. We studied the pole motion of rational solution of the BKP and other hierarchies which can be handled explicitly. In particular, for BKP rational solution we obtained a matrix pair X,Y which can be regarded as the Banalogue of the Moser pair, and represented its tau function as the Pfaffian of timedependent linear combination of Y and powers of X. We also studied, jointly with A.Yu.Orlov, solutions of hypergeometric type to various soliton equations, and interpreted them as discrete analogues (ones with integrals replaced by sums) of various matrix models. For this we expanded the partition function of normal matrix model by Schur functions (hence rediscovered that the partition functions are Toda tau functions, and specialized the time variables in an appropriate way to obtain a different representation of it in terms of the sum over partitions, and interpreted the resulting formulae as discrete versions of various matrix models. This way we obtained discrete versions of normal, Hermite, and unitary matrix models.

Research Products
(2 results)