Riemann-Hilbert approach and N Double-Pole Solutions for a nonlinear Schrödinger-type equation

In this paper, the inverse scattering transform for the Schrödinger-type equation is studied with zero boundary conditions (ZBCs) via the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. And then the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving RH problems corresponding to the reflectionless cases. Furthermore, we present single double-pole solution by taking some parameters, which is analyzed in detail.