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We consider a spatially non-autonomous discrete sine-Gordon equation with constant forcing and its continuum limit(s) to model a 0-π Josephson junction with an applied bias current. The continuum limits correspond to the strong coupling limit of the discrete system. The nonautonomous character is due to the presence of a discontinuity point, namely a jump of π in the sine-Gordon phase. The continuum models admits static solitary waves which are called π-kinks and are attached to thedoi:10.1137/test5 fatcat:skqkrackcbej5ac5dkmpvqjrly