THE MINIMAL N=2 SUPEREXTENSION OF THE NLS EQUATION

TL;DR

This paper reveals that the N=1 nonlinear Schrödinger (NLS) equation actually has hidden N=2 supersymmetry, which becomes clear when reformulated with N=2 superfields, linking it to the N=2 KdV hierarchy and superconformal algebra.

Contribution

The paper demonstrates the N=2 supersymmetry of the N=1 NLS equation and introduces a KP-like Lax operator in N=2 superfields that captures the hierarchy's conserved currents.

Findings

N=2 supersymmetry of the N=1 NLS equation is established.

The second Hamiltonian structure matches the N=2 superconformal algebra.

A new Lax operator reproduces all conserved currents of the hierarchy.

Abstract

We show that the well known $N=1$ NLS equation possesses $N=2$ supersymmetry and thus it is actually the $N=2$ NLS equation. This supersymmetry is hidden in terms of the commonly used $N=1$ superfields but it becomes manifest after passing to the $N=2$ ones. In terms of the new defined variables the second Hamiltonian structure of the supersymmetric NLS equation coincides with the $N=2$ superconformal algebra and the $N=2$ NLS equation belongs to the $N=2$ $a=4$ KdV hierarchy. We propose the KP-like Lax operator in terms of the $N=2$ superfields which reproduces all the conserved currents for the corresponding hierarchy.