Integrable Discretisations for a Class of Nonlinear Schrodinger Equations on Grassmann Algebras

TL;DR

This paper develops integrable discretisations of coupled nonlinear Schrödinger equations with Grassmann algebra entries, introducing Darboux transformations and deriving Grassmann generalisations of Toda and NLS systems with discrete Lax representations.

Contribution

It presents novel integrable discretisations for Grassmann algebra-based NLS equations, including Darboux transformations and discrete Lax pairs, extending classical integrable systems.

Findings

Grassmann generalisations of Toda lattice and NLS dressing chain

Discrete Lax representations for the new systems

Formulation of initial value and boundary problems

Abstract

Integrable discretisations for a class of coupled (super) nonlinear Schrodinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initialboundary problems are formulated.