Supersymmetric extensions of the nonlinear Schrödinger equation: Symmetries and coverings

Abstract

A construction is proposed for a supersymmetric generalization of the cubic Schrödinger equation, resulting in two supersymmetric systems, one of which contains a free parameter. Both systems are proven to admit an infinite set of (higher-order) local and nonlocal symmetries and a seemingly infinite set of conservation laws, the lowest-order terms of which are given explicitly. Moreover, the theory of coverings (equivalent to the prolongation method of Wahlquist and Estabrook) is applied to both systems. Both are seen to admit an infinite-dimensional covering algebra, the structure of which is determined explicitly, resulting in a related supersymmetric system of differential equations, as well as an auto-Bäcklund transformation for each equation. This indicates the complete integrability of both systems.