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

TL;DR

This paper demonstrates that hypergeometric KP tau functions can be viewed as infinite-soliton solutions of dual multi-component KP hierarchies, revealing a novel duality and explicit soliton parameter relations.

Contribution

It establishes a duality between hypergeometric tau functions and multi-component KP hierarchies, providing explicit soliton solutions and parameter relations.

Findings

Hypergeometric tau functions are infinite-soliton solutions of dual hierarchies.

When polynomial, they yield N-soliton solutions with parameters linked to partitions.

The roles of variables are interchanged in the dual hierarchy representation.

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 $\tau({\bf t},T,{\bf t}^*)$.