Integrable discretisations of the noncommutative NLS equation

Abstract

We show how to derive noncommutative versions of integrable partial difference equations using Darboux transformations. As an illustrative example, we use the nonlinear Schr\"odinger (NLS) system. We derive a noncommutative nonlinear Schr\"odinger equation and we construct its integrable discretisations via the compatibility condition of Darboux transformations around the square. In particular, we construct a noncommutative Adler--Yamilov type system and a noncommutative discrete Toda equation. For the noncommutative Adler--Yamilov type system we construct B\"acklund transformations.