Asymptotic Behavior of Solutions to the Nonlinear Schrödinger Equation with Non-Zero Boundary Conditions in the Presence of a Pair of Second-Order Discrete Spectra

The nonlinear Schrödinger equation is a classical nonlinear evolution equation with wide applications. This paper explores the asymptotic behavior of solutions to the nonlinear Schrödinger equation with non-zero boundary conditions in the presence of a pair of second-order discrete spectra. We analyze the Riemann–Hilbert problem in the inverse scattering transform by the Deift–Zhou nonlinear steepest descent method. Then we propose a proper deformation to deal with the growing time term and give the conditions for the series in the process of deformation by the Laurent expansion. Finally, we provide the characterization of the interactions between the solitary waves corresponding to second-order discrete spectra and the coherent oscillations produced by the perturbation. Numerical verifications are also performed.