Some problems for a system of nonlinear evolution equations on a half-line

This chapter aims to apply the results of the inverse scattering problems (ISPs) to solving an initial value problem and some initial-boundary value problems (IBVPs) for the system of nonlinear evolution equations (NLEEs). The IBVPs for the considered system of NLEEs and the attractive nonlinear Schrodinger (NLS) equation are solved by applying the results of the ISP for a system of first-order ordinary differential equations (ODEs) on a half-line with a potential non-self-adjoint matrix. The chapter seeks to find exact solutions of the considered NLEEs in the class of non-scattering potentials. It also aims to apply the results of the ISP for the system of first-order ODEs on a half-line with a potential self-adjoint matrix to solving the Cauchy initial-value for the repulsive NLS equation.