Riemann-Hilbert approach for the NLSLab equation with nonzero boundary conditions

Abstract

We consider the inverse scattering transform for the nonlinear Schr\"{o}dinger equation in laboratory coordinates (NLSLab equation) with nonzero boundary conditions (NZBCs) at infinity. In order to better deal with the scattering problem of NZBCs, we introduce the two-sheeted Riemann surface of $\kappa$, then it convert into the standard complex $z$-plane. In the direct scattering problem, we study the analyticity, symmetries and asymptotic behaviors of the Jost function and the scattering matrix in detail. In addition, we establish the discrete spectrum, residual conditions, trace foumulae and theta conditions for the case of simple poles and double poles. The inverse problems of simple poles and double poles are from the Riemann-Hilbert problem (RHP). Finally, we obtain some soliton solutions of the NLSLab equation, including stationary solitons, non-stationary solitons and multi-soliton solutions. Some features of these soliton solutions caused by the influences of each parameters are analyzed graphically in order to control such nonlinear phenomena.