Riemann-Hilbert approach for the integrable discrete Hirota equation with bounded boundary conditions in the presence of a discrete spectrum

In this paper, the Riemann-Hilbert (RH) approach for the integrable discrete Hirota equation with bounded boundary conditions is presented. In the direct scattering problem, we study the analyticity, asymptotics, symmetries of the eigenfunctions, and scattering coefficients and analyze the distribution of discrete eigenvalues. In the inverse scattering problem, the RH problem is constructed and solved as well as the reconstruction formula of potential is derived based on asymptotics. Finally, combining the time evolution, we solve the first- and second-order dark soliton solutions on the nonzero background under the reflectionless condition.