The Extended Supersymmetrization of the Multicomponent Kadomtsev--Petviashvilli Hierarchy

TL;DR

This paper presents three methods for extending the multicomponent KP hierarchy with N=2 supersymmetry, showing how soliton equations can be embedded and generalized within supersymmetric frameworks.

Contribution

It introduces three distinct approaches to supersymmetrize the multicomponent KP hierarchy, analyzing their effects on soliton equations and their bosonic limits.

Findings

Many soliton equations can be embedded using the first approach without altering their bosonic form.

The second and third approaches produce generalized soliton equations within the usual commuting function class.

The bosonic part of the one-component supersymmetric KP hierarchy matches the classical two-component KP hierarchy.

Abstract

We describe three different approaches to the extended (N=2) supersymmetrization of the multicomponent KP hierarchy. In the first one we utilize only superfermions while in the second only superbosons and in the third superbosons as well as superfermions. It is shown that many soliton equations can be embedded in the supersymmetry theory by using the first approach even if we do not change these equations in the bosonic limit of the supersymmetry. In the second or third approach we obtain a generalization of the soliton equations in the bosonic limit which remains in the class of the usual commuting functions. As the byproduct of our analysis we prove that for the first procedure the bosonic part of the one--component supersym\-me\-tric KP hierarchy coincides with the usual classical two--component KP hierarchy.