Integration of loaded nonlinear Schrödinger equation in class of fast decaying functions

We show that the inverse scattering transform technique can be applied to obtain the time dependence of scattering data of the Zakharov - Shabat system, which is described by the loaded nonlinear Schrödinger equation in the class of fast decaying functions. In addition we prove that the Cauchy problem for the loaded nonlinear Schrödinger equation is uniquely solvable in the class of rapidly decreasing functions. We provide the explicit expression of a single soliton solution for the loaded nonlinear Schrödinger equation. As an example, we find the soliton solution of the considered problem for an arbitrary non - zero continuous function $\gamma(t).$