  A system of linear integral equations is presented, which is the analog of
the system of Marchenko integral equations, to solve the inverse scattering
problem for the linear system associated with the derivative NLS equations. The
corresponding direct and inverse scattering problems are analyzed, and the
recovery of the potentials and the Jost solutions from the solution to the
Marchenko system is described. When the reflection coefficients are zero, some
explicit solution formulas are provided for the potentials and the Jost
solutions in terms of a pair of constant matrix triplets representing the
bound-state information for any number of bound states and any multiplicities.
In the reduced case, when the two potentials in the linear system are related
to each other through complex conjugation, the corresponding reduced Marchenko
integral equation is obtained. The solution to the derivative NLS equation is
obtained from the solution to the reduced Marchenko integral equation. The
theory presented is illustrated with some explicit examples.
