New integrable equations of nonlinear Schrodinger type

New integrable matrix nonlinear evolution partial differential equations in (1 + 1)-dimensions are derived, via a treatment which starts from an appropriate matrix generalization of the Zakharov-Shabat spectral problem. Via appropriate parametrizations, multi-vector versions of these equations are also exhibited. Generally these equations feature solitons that do not move with constant velocities: they rather behave as boomerons or as trappons, namely, up to a Galilelan transformation, they typically boomerang back to where they came from, or they are trapped to oscillate around some fixed position determined by their initial data. In this paper, meant to be the first of a series, we focus on the derivation and exhibition of new coupled evolution equations of nonlinear Schrodinger type and on the behavior of their single-soliton solutions.