Revisiting B\"acklund-Darboux transformations for KP and BKP integrable hierarchies

TL;DR

This paper explores Bäcklund-Darboux transformations for KP, BKP, and related hierarchies using bilinear formalism, extending to discrete versions and employing operator methods from the Kyoto school.

Contribution

It extends Bäcklund-Darboux transformation theory to discrete integrable hierarchies and integrates operator approaches with tau-function formalism.

Findings

Extended Bäcklund-Darboux transformations to fully discrete hierarchies.

Unified bilinear formalism for continuous and discrete integrable equations.

Applied operator approach to construct transformations using fermionic fields.

Abstract

We consider B\"acklund-Darboux transformations for integrable hierarchies of nonlinear equations such as KP, BKP and their close relatives referred to as modified KP and Schwarzian KP. We work in the framework of the bilinear formalism based on the bilinear equations for the tau-function. This approach allows one to extend the theory to fully difference (or discrete) versions of the integrable equations and their hierarchies in a natural way. We also show how to construct the B\"acklund-Darboux transformations in the operator approach developed by the Kyoto school, in which the tau-functions are represented as vacuum expectation values of certain operators made of free fermionic fields (charged for KP and neutral for BKP).