The algebraic structure behind the derivative nonlinear Schroedinger equation

TL;DR

This paper develops an algebraic framework for the Kaup-Newell hierarchy, including the derivative nonlinear Schrödinger equation, using affine algebraic structures, and constructs soliton solutions systematically.

Contribution

It introduces a higher grading affine algebraic construction for the Kaup-Newell hierarchy and demonstrates the algebraic derivation of integrable models and their solutions.

Findings

Derived the Kaup-Newell hierarchy using affine algebraic methods.

Constructed soliton solutions for the hierarchy systematically.

Showed the equivalence of different spectral problems via automorphism.

Abstract

The Kaup-Newell (KN) hierarchy contains the derivative nonlinear Schr\" odinger equation (DNLSE) amongst others interesting and important nonlinear integrable equations. In this paper, a general higher grading affine algebraic construction of integrable hierarchies is proposed and the KN hierarchy is established in terms of a $\hat{s\ell}_2$ Kac-Moody algebra and principal gradation. In this form, our spectral problem is linear in the spectral parameter. The positive and negative flows are derived, showing that some interesting physical models arise from the same algebraic structure. For instance, the DNLSE is obtained as the second positive, while the Mikhailov model as the first negative flows, respectively. The equivalence between the latter and the massive Thirring model is explicitly demonstrated also. The algebraic dressing method is employed to construct soliton solutions in a systematic manner for all members of the hierarchy. Finally, the equivalence of the spectral problem introduced in this paper with the usual one, which is quadratic in the spectral parameter, is achieved by setting a particular automorphism of the affine algebra, which maps the homogeneous into principal gradation.