Explorations of the Extended ncKP Hierarchy

TL;DR

This paper explores an extended noncommutative KP hierarchy, deriving formulas, analyzing reductions, extending Sato formalism, and demonstrating that soliton solutions persist in the extended framework.

Contribution

It introduces an extension of the ncKP hierarchy with new evolution equations, extending the Sato formalism and analyzing soliton solutions within this generalized setting.

Findings

Derived efficient formulas for the extended hierarchy equations

Extended the bilinear identity theorem to the generalized framework

Showed N-soliton solutions remain valid in the extended hierarchy

Abstract

A recently obtained extension (xncKP) of the Moyal-deformed KP hierarchy (ncKP hierarchy) by a set of evolution equations in the Moyal-deformation parameters is further explored. Formulae are derived to compute these equations efficiently. Reductions of the xncKP hierarchy are treated, in particular to the extended ncKdV and ncBoussinesq hierarchies. Furthermore, a good part of the Sato formalism for the KP hierarchy is carried over to the generalized framework. In particular, the well-known bilinear identity theorem for the KP hierarchy, expressed in terms of the (formal) Baker-Akhiezer function, extends to the xncKP hierarchy. Moreover, it is demonstrated that N-soliton solutions of the ncKP equation are also solutions of the first few deformation equations. This is shown to be related to the existence of certain families of algebraic identities.