The paper deals with Cauchy problems for the system of non-linear equations on a half-line ic_1t=c_1xx-2c₁²c₂ -ic_2t=c_2xx-2c₁c₂² 0<or=x< infinity . The first problem for the non-linear Schrodinger equation (NLS) is obtained from the system when c₁=C₂. In this case the boundary problem associated with this NLS does not have a discrete spectrum, and the solution of the NLS is found uniquely from the given initial condition. Further we show a remarkable class of matrix potentials, in which the inverse scattering problem associated with the system can be solved exactly. In the case that the given scattering data for consist of only two simple poles the exact soliton solution of the system is presented. This happens if and only if the time evolution of the scattering data for obeys some time evolution equations. In this case the Cauchy problem for the NLS, which is obtained when c₁=-c₂, is solved exactly.