ABSTRACT
In this chapter, the authors aim to apply results of the inverse scattering problem (ISP) for the system of first-order ordinary differential equations. They show that every scattering data set of the associated scattering problem corresponds a unique solution of the initial-boundary value problems for the modified Korteweg-de Vries equations on the half-line. The authors also show the ISP, which is to determine the potential matrix from the scattering data, and also to describe the scattering data, that is, to establish necessary and sufficient conditions for given quantities to be scattering data for the system on a half-line with boundary condition. The scattering problem with the non-scattering potentials is completely determined by the characteristics of the discrete spectrum.
