The determinant representation of the gauge transformation for the discrete KP hierarchy

TL;DR

This paper develops a determinant representation for gauge transformations in the discrete KP hierarchy, enabling the construction of new tau functions via generalized discrete Wronskian determinants.

Contribution

It introduces a novel determinant representation of gauge transformation operators for the discrete KP hierarchy and constructs new tau functions using generalized discrete Wronskian determinants.

Findings

Established the determinant form of the gauge transformation operator.

Derived a new tau function from the initial tau using the determinant representation.

Introduced properties of the discrete difference operator and generalized Wronskian determinants.

Abstract

A successive gauge transformation operator $T_{n+k}$ for the discrete KP(dKP) hierarchy is defined, which is involved with two types of gauge transformations operators. The determinant representation of the $T_{n+k}$ is established,and then it is used to get a new $tau$ function $\tau^{(n+k)}_\triangle$ of the dKP hierarchy from an initial $\tau_\triangle$. In this process, we introduce a generalized discrete Wronskian determinant and some useful properties of discrete difference operator.