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In this article we formulate the direct and inverse scattering theory for the focusing matrix Zakharov-Shabat system as the construction of a 1, 1-correspondence between focusing potentials with entries in $L^1(\mathbb{R})$ and Marchenko integral kernels, given the fact that these kernels encode the usual scattering data (one reflection coecient, the discrete eigenvalues with positive imaginary part, and the corresponding norming constants) faithfully. In the re ectionless case, we solve the Marchenko equations explicitly using matrix triplets and obtain focusing matrix NLS solutions in closed form.

Novel formulation of inverse scattering and characterization of scattering data