The direct and inverse scattering problems on the full line are analyzed for a first-order system of ordinary linear differential equations associated with the derivative nonlinear Schrödinger equation and related equations. The system contains a spectral parameter and two potentials, where the potentials are proportional to the spectral parameter and hence are called energy-dependent potentials. Using the two potentials as input, the direct problem is solved by determining the scattering coefficients and the bound-state information consisting of bound-state energies, their multiplicities, and the corresponding norming constants. By using two different methods, the corresponding inverse problem is solved by determining the two potentials when the scattering data set is used as input. The first method involves the transformation of the energy-dependent system into two distinct energy-independent systems. The second method involves the establishment of the so-called alternate Marchenko system of linear integral equations and the recovery of the energy-dependent potentials from the solution to the alternate Marchenko system.

The direct and inverse scattering problems are analyzed for a first-order discrete system associated with the semidiscrete version of the derivative nonlinear Schrödinger (NLS) system. The Jost solutions, the scattering coefficients, the bound-state dependency and norming constants are investigated and related to the corresponding quantities for two particular discrete linear systems associated with the semi-discrete version of the NLS system. The bound-state data set with any multiplicities is described in an elegant manner in terms of a pair of constant matrix triplets. Several methods are presented to solve the inverse problem to recover the potential values in the first-order discrete system. One of these methods uses a newly derived, standard discrete Marchenko system using as input the scattering data directly coming from the first-order discrete system. This new Marchenko method is presented in a way that it is generalizable to other first-order systems both in the discrete and continuous cases for which a Marchenko system and a Marchenko theory are not yet available. Finally, using the time-evolved scattering data set, the inverse scattering transform is applied on the corresponding semi-discrete derivative NLS system, and in

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.

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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 DNLS (derivative nonlinear Schrödinger) 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 DNLS equation is obtained from the solution to the reduced Marchenko integral equation. The theory presented is illustrated with some explicit examples.