Inverse scattering problems for systems of first-order ODEs on a half-line

This chapter discusses two inverse scattering problems on a half-line for systems of first-order ordinary differential equations with a non-self-adjoint and a self-adjoint potential matrices. The self-adjoint problem does not have a discrete spectrum. It is associated mostly with a Cauchy initial value problem for integrable nonlinear equations (NLEEs), whereas the non-self-adjoint problem is associated mostly with an initial-boundary value problem for integrable NLEEs. The chapter also discusses the connection between the analytic solution and Jost solutions. It describes the main theorem regarding the complete description of the scattering function, that is the establishment of necessary and sufficient conditions for a given function to be the scattering function for system on a half-line with a potential self-adjoint matrix and with boundary condition.