Tau-functions of integrable discrete massive Thirring model under nonvanishing boundary conditions

The connection between the massive Thirring (MT) model and the KP hierarchy is made clear recently and it is shown that the MT model can be derived from a set of four bilinear equations under nonvanishing boundary conditions. In the present work, we demonstrate that all four bilinear equations originate from a set of four discrete KP equations, thus the integrable discretizations of these four bilinear equations are naturally given. Based on this finding, the semi- and fully discrete MT model equations are re-derived, the dark soliton, breather and first-order rogue wave solutions are constructed.