Soliton equations and combinatories

众所周知，KP有理解的极点运动服从Calogero-Moser动力系统族(类似地，KP族的三角解和椭圆解也是如此)。在J.F.Van Diejen最近发现KdV波函数的零点与Ruijsenaars-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。从李氏理论的观点来看，KP和Calogero-Moser系统都允许具有各种内部对称性的推广。我们研究了BKP和其他可以显式处理的方程组的有理解的极点运动。特别地，对于BKP有理解，我们得到了一个可视为Moser对的B-模拟的矩阵对X，Y，并将其tau函数表示为Y与X的幂的含时线性组合的Pfaffian。我们还与A.Yu.Orlov共同研究了各种孤子方程的超几何形式的解，并将它们解释为各种矩阵模型的离散模拟(用和代替积分)。为此，我们用Schur函数对正态矩阵模型的配分函数进行了扩展(从而重新发现配分函数是Toda-tau函数，并对时间变量进行适当的特化，得到了它在分拆上的和的不同表示，并将所得公式解释为各种矩阵模型的离散形式。通过这种方法，我们得到了正规、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.