Riemann–Hilbert approach and $N$-soliton solutions of the generalized mixed nonlinear Schrödinger equation with nonzero boundary conditions

We apply the inverse scattering transformation to the generalized mixed nonlinear Schrödinger equation with nonzero boundary condition at infinity. The scattering theories are investigated. In the direct problem, we analyze the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and the scattering matrix, and the properties of the discrete spectrum. In the inverse problem, an appropriate Riemann–Hilbert problem is formulated. By solving the problem, we obtain the reconstruction formula, the trace formula, and the “theta” condition. In the reflectionless case, a complicated integral factor is derived, which is a key ingredient of the explicit expression for $N$-soliton solutions. Using the $N$-soliton formula, we discuss the abundant dynamical features of the solution and its phases at different parameter values.