Hypergeometric tau functions $\tau({\bf t},T,{\bf t}^*)$ as $\infty$-soliton tau function in T variables

Abstract

We consider KP tau function of hypergeometric type $\tau({\bf t},T,{\bf t}^*)$, where the set ${\bf t}$ is the KP higher times and $T,{\bf t}^*$ are sets of parameters. Fixing ${\bf t}^*$, we find that $\tau({\bf t},T,{\bf t}^*)$ is an infinite-soliton solution of different (dual) multi-component KP (and TL) hierarchy, where the roles of the variables ${\bf t}$ and $T$ are interchanged. When $\tau({\bf t},T,{\bf t}^*)$ is a polynomial in ${\bf t}$, we obtain a $N$-soliton solution of the dual hierarchy. Parameters of the solitons are related to the Frobenius coordinates of partitions in the Schur function development of $\tau({\bf t},T,{\bf t}^*)$.