On Darboux-B\"acklund Transformations for the Q-Deformed Korteweg-de Vries Hierarchy

TL;DR

This paper explores Darboux-Bäcklund transformations within the q-deformed Korteweg-de Vries hierarchy, introducing new elementary transformations and representations of the tau-function using q-deformed pseudodifferential operators.

Contribution

It identifies elementary DBTs triggered by gauge operators and develops q-deformed Wronskian and binary-type tau-function representations.

Findings

Elementary DBTs are constructed from gauge operators.

Iterated DBTs yield q-deformed Wronskian and binary tau-function representations.

The approach advances understanding of q-deformed integrable hierarchies.

Abstract

We study Darboux-B\"acklund transformations (DBTs) for the $q$-deformed Korteweg-de Vries hierarchy by using the $q$-deformed pseudodifferential operators. We identify the elementary DBTs which are triggered by the gauge operators constructed from the (adjoint) wave functions of the hierarchy. Iterating these elementary DBTs we obtain not only $q$-deformed Wronskian-type but also binary-type representations of the tau-function to the hierarchy.