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Using a bidifferential graded algebra approach to integrable partialdifferential or difference equations, a unified treatment of continuous,semi-discrete (Ablowitz-Ladik) and fully discrete matrix NLS systems ispresented. These equations originate from a universal equation within thisframework, by specifying a representation of the bidifferential graded algebraand imposing a reduction. By application of a general result, correspondingfamilies of exact solutions are obtained that in particular comprise the matrixsoliton solutions in the focusing NLS case. The solutions are parametrised interms of constant matrix data subject to a Sylvester equation (which previouslyappeared as a rank condition in the integrable systems literature). These dataexhibit a certain redundancy, which we diminish to a large extent.  More precisely, we first consider more general AKNS-type systems from whichtwo different matrix NLS systems emerge via reductions. In the continuous case,the familiar Hermitian conjugation reduction leads to a continuous matrix(including vector) NLS equation, but it is well-known that this does not workas well in the discrete cases. On the other hand there is a complex conjugationreduction, which apparently has not been studied previously. It leads to squarematrix NLS systems, but works in all three cases (continuous, semi- andfully-discrete). A large part of this work is devoted to an exploration of thecorresponding solutions, in particular regularity and asymptotic behaviour ofmatrix soliton solutions.

Solutions of matrix NLS systems and their discretisations: A unified treatment