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

Abstract

We consider KP tau function of hypergeometric type $\tau (\mathbf{t},T,{\mathbf{t}}^{\ast })$, where the set $\mathbf{t}$ is the KP higher times and $T,{\mathbf{t}}^{\ast }$ are sets of parameters. Fixing ${\mathbf{t}}^{\ast }$, we find that $\tau (\mathbf{t},T,{\mathbf{t}}^{\ast })$ is an infinite-soliton solution of different (dual) multi-component KP (and TL) hierarchy, where the roles of the variables $\mathbf{t}$ and $T$ are interchanged. When $\tau (\mathbf{t},T,{\mathbf{t}}^{\ast })$ is a polynomial in $\mathbf{t}$, 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 $\tau (\mathbf{t},T,{\mathbf{t}}^{\ast })$.

Latex, 24 pages, no figures