Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras

• Elementary Darboux transforms for Grassmann-extended NLS equations are constructed.
• Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained.
• Grassmann generalisations of the difference Toda and NLS equations are obtained.
• For these systems initial value and initial-boundary problems are formulated.

Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (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 initial-boundary problems are formulated.