Riemann–Hilbert approach for the inhomogeneous discrete nonlinear Schrödinger equation with nonzero boundary conditions

In this paper, we systematically investigate the Riemann–Hilbert (RH) approach and obtain the soliton solutions for the inhomogeneous discrete nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBCs). Starting from the spectral problem and introducing the uniformization variable $\kappa$ to avoid the complexity of double-valued function and Riemann surface, we deduce the analyticity, asymptotics and symmetries of the eigenfunctions and scattering coefficients, then the RH problem and reconstruction formula for the potential are successfully constructed. Under reflectionless condition and combining the time evolution of the scattering coefficients and eigenfunctions, we obtain various first-order soliton solutions with different direction of propagation caused by the change of the coefficients. Based on the analytic solution and the choice of special parameter values, we obtain the collision mechanism of two soliton solutions. Furthermore, the important advantage of the RH problem is that it can be further used to study the soliton resolution and the long-time asymptotic behavior of the solutions.