Soliton equations and combinatories

众所周知，KP有理解的极点运动服从卡罗伽罗-莫泽动力系统的层次（同样适用于KP层次的三角解和椭圆解）。在J.F.Van Diejen最近发现KdV波函数的零点与rujsenaars - schneider系统之间的关系后，这一事实可能会重新引起人们的兴趣。从李论的观点来看，KP和伽罗伽罗-莫泽层次都允许具有各种内部对称性的推广。研究了BKP的有理解的极点运动和其他可显式处理的层次。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，并将其tau函数表示为Y与X的幂次的随时间线性组合的Pfaffian。Orlov，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（用和代替积分的类似物）。为此，我们用Schur函数展开了普通矩阵模型的配分函数（从而重新发现配分函数是Toda - tau函数），并对时间变量进行了适当的专门化，得到了它的不同的分区和表示形式，并将所得公式解释为各种矩阵模型的离散版本。通过这种方法，我们得到了正态、Hermite和酉矩阵模型的离散版本。

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.