A unified approach to the AKNS, DNLS, KP and mKP hierarchies in the anti-self-dual Yang-Mills reduction

TL;DR

This paper presents a unified framework connecting various integrable hierarchies, including AKNS, DNLS, KP, and mKP, through the anti-self-dual Yang-Mills reduction, and provides explicit solutions using quasi-determinants.

Contribution

It introduces a comprehensive approach unifying multiple integrable hierarchies within the ASDYM reduction and derives explicit solutions in a systematic way.

Findings

Unified formulation of AKNS, DNLS, KP, and mKP hierarchies.

Explicit Gram-type solutions expressed via quasi-determinants.

Connection between bilinearization and ASDYM reduction.

Abstract

We show a unified approach to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and the unreduced derivative nonlinear Schr\"odinger (DNLS) hierarchies (including the Kaup-Newell, Chen-Lee-Liu, Gerdjikov-Ivanov and a generalized DNLS), together with their multi-component extensions, in the framework of the anti-self-dual Yang-Mills (ASDYM) reduction. By restricting the gauge group to GL(2), the Kadomtsev-Petviashvili (KP) and modified KP (mKP) hierarchies are formulated in the ASDYM reduction via squared eigenfunction symmetry constraints. In this case, the bilinearization of the generalized DNLS equations can also be understood through this reduction. Finally, Gram-type exact solutions for the relevant equations are presented in terms of quasi-determinants.