Constructing Soliton Solutions of the Nonlinear Schrödinger Equation by Inverse Scattering and Hirota’s Direct Methods Pham Loi Vu & Nguyen Huy Hoang Abstract            We construct the exact complex-valued solutions of the nonlinear Schrodinger equation (NLS) in the class of nonscattering potentials, where the inverse problem associated with the NLS equation can be solved exactly. It is shown that in this class if  the solution of the inverse problem satisfies the NLS equation and if the singular numbers of this problem satisfy some conditions, then ths degree of normalization polynomials generated by the discrete spectrum must be zero and the polynomials are reduced to the corresponding normalization factors, which depend on time only. If the degree of normalization polynomials is zero, then the general $N$-soliton solution $q$ of the NLS equation is given by the transform $q=F/G$, where $F$ and $G$ are representd in the explicit forms in terms of the given scatterin data.