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Soliton equations and combinatories

孤子方程和组合

基本信息

批准号：
15540165
负责人：
SHIOTA Takahiro
金额：
$ 1.54万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

As is well-known, the motion of poles of a KP rational solution obeys the hierarchy of Calogero-Moser 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 Ruijsenaars-Schneider system. Both the KP and the Calogero-Moser 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 B-analogue of the Moser pair, and represented its tau function as the Pfaffian of time-dependent 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.

众所周知，KP有理解的极点的运动服从Calogero-Moser动力系统的族（KP族的三角解和椭圆解也是如此）。在J.F.货车Diejen最近发现KdV波函数的零点与Ruijsenaars-Schneider系统之间的关系之后，这一事实可能会重新引起人们的兴趣。KP和Calogero-Moser层次结构都允许从李理论的角度具有各种内部对称性的推广。研究了BKP方程及其它可显式处理的方程族的有理解的极点运动。特别地，对于BKP有理解，我们得到了一个矩阵对X，Y，它可以看作是Moser对的B-模拟，并把它的τ函数表示为Y与X的幂的依赖于时间的线性组合的Pfiran。我们还与A.Yu.奥尔洛夫一起研究了各种孤子方程的超几何型解，并将它们解释为各种矩阵模型的离散类似物（用和代替积分）。为此，我们用Schur函数扩展了正规矩阵模型的配分函数（从而重新发现配分函数是户田τ函数），并以适当的方式对时间变量进行专门化，以获得它在分块求和方面的不同表示，并将所得公式解释为各种矩阵模型的离散版本。通过这种方式，我们获得了离散形式的正常，埃尔米特和酉矩阵模型。