Riemann–Hilbert approach and N double-pole solutions for a nonlinear Schr?dinger-type equation

We investigate the inverse scattering transform for the Schro¨dinger-type equation under zero boundary conditions with 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. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schr?dinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.