The tau functions of the constrained CKP hierarchy

TL;DR

This paper constructs tau functions for the constrained CKP hierarchy using CKP Darboux transformations, advancing understanding of its special tau function structure within the KP hierarchy.

Contribution

It introduces a method to explicitly construct tau functions for the constrained CKP hierarchy via Darboux transformations.

Findings

Explicit tau functions for the constrained CKP hierarchy are obtained.

The construction applies to both odd and even positive integers k.

Enhances the theoretical framework of the CKP hierarchy and its tau functions.

Abstract

The CKP hierarchy is one important sub-hierarchy of the KP hierarchy, which is quite special due to its tau function. Here we construct the tau functions for the constrained CKP hierarchy $(L^k)_{<0}=\sum_{i=1}^{m}\big(q_{1,i}\partial^{-1}q_{2,i}-(-1)^kq_{2,i}\partial^{-1}q_{1,i}\big)$ with $k$ being odd or even positive integer by using the CKP Darboux transformations.