Noncommutative NLS systems: Darboux--Bäcklund transformations and integrable discretisations

Abstract

We study noncommutative analogues and integrable discretisations of nonlinear Schrödinger (NLS)-type systems associated with reduction groups. In particular, we consider the Ablowitz--Kaup--Newell--Segur (AKNS) system, the Kaup--Newell derivative NLS system, and the Mikhailov--Shabat--Yamilov deformation of the derivative NLS system together with their Darboux--Bäcklund transformations and associated lattice equations. We derive the continuum limits of previously constructed integrable lattice systems and recover the corresponding NLS-type partial differential equations. We then construct a noncommutative deformation of the Mikhailov--Shabat--Yamilov system and show that, unlike the AKNS and Kaup--Newell cases, its Lax representation requires the introduction of nonlocal variables. Furthermore, we derive Darboux--Bäcklund transformations and integrable discretisations for the noncommutative derivative NLS and deformation derivative NLS systems in the form of vertex--bond lattice equations. We also construct explicit solutions for a six-point derivative NLS-type lattice equation and for its noncommutative analogue.