Tau Functions of (n,1) curves and Soliton Solutions on Non-Zero Constant Backgrounds

TL;DR

This paper investigates tau functions of (n,1) curves within the KP-hierarchy, revealing their behavior and generating soliton solutions on non-zero constant backgrounds through vertex operators.

Contribution

It introduces a detailed analysis of tau functions for (n,1) curves and constructs soliton solutions on non-zero backgrounds using vertex operators.

Findings

Tau functions are trivial when α=0.

Tau functions become exponential of quadratic functions when α≠0.

Vertex operators generate soliton solutions on non-zero 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.