B\"acklund transformations between the AKNS and DNLS hierarchies

TL;DR

This paper derives Bäcklund transformations linking solutions of the AKNS and DNLS hierarchies, revealing their deep connections and shared tau-functions, and relating them to the Ablowitz-Ladik hierarchy.

Contribution

It introduces a set of Bäcklund transformations between AKNS and DNLS hierarchies using Miura-like transformations and shows their common tau-functions and relation to Ablowitz-Ladik hierarchy.

Findings

AKNS and DNLS hierarchies are connected via Bäcklund transformations.

Both hierarchies share a common set of tau-functions.

The hierarchies are related to the Ablowitz-Ladik hierarchy.

Abstract

Starting from the functional representation of the Ablowitz-Kaup-Newell-Segur (AKNS) and derivative nonlinear Schr\"odinger (DNLS) hierarchies and using the chains of the Miura-like transformations we derive a set of B\"acklund transformations that link solutions of these systems. It is shown that the extended AKNS and DNLS hierarchies possess common set of tau-functions and their connection with the Ablowitz-Ladik hierarchy is established. These results are another manifestation of the already known fact that the AKNS and DNLS hierarchies are closely related and can be viewed as particular cases of a more general system.