Tau functions of (n, 1) curves and soliton solutions on nonzero constant backgrounds

Abstract

We study the tau function of the KP-hierarchy associated with an (n,1) curve $y^n=x-\alpha$. If $\alpha=0$ the corresponding tau function is 1. On the other hand if $\alpha\neq 0$ the tau function becomes the exponential of a quadratic function of the time variables. By applying vertex opertaors to the latter we obtain soliton solutions on non-zero constant backgrounds.