Darboux Transformation for the Non-isospectral AKNS Hierarchy and Its Asymptotic Property

TL;DR

This paper develops a Darboux transformation for the non-isospectral AKNS hierarchy, revealing its non-auto-Bäcklund nature and providing a method to derive soliton solutions for related nonlinear equations.

Contribution

It introduces a novel Darboux transformation for the non-isospectral AKNS hierarchy and details how it alters integral constants, enabling soliton solution derivation.

Findings

Darboux transformation for non-isospectral AKNS hierarchy constructed

Transformation changes integral constants, not an auto-Bäcklund transformation

Soliton solutions for derived nonlinear equations obtained

Abstract

In this article, the Darboux transformation for the non-isospectral AKNS hierarchy is constructed. We show that the Darboux transformation for the non-isospectral AKNS hierarchy is not an auto-B\"acklund transformation, because the integral constants of the hierarchy will be changed after the transformation. The transform rule of the integral constants will be also derived. By this means, the soliton solutions of the nonlinear equations derived by the non-isospectral AKNS hierarchy can be found.