Stability analysis of pi-kinks in a 0-pi Josephson junction

We consider a spatially non-autonomous discrete sine-Gordon equation with constant forcing and its continuum limit(s) to model a 0-pi Josephson junction with an applied bias current. The continuum limits correspond to the strong coupling limit of the discrete system. The non-autonomous character is due to the presence of a discontinuity point, namely a jump of pi in the sine-Gordon phase. The continuum models admits static solitary waves which are called pi-kinks and are attached to the discontinuity point. For small forcing, there are three types of pi-kinks. We show that one of the solitary waves is stable and the others are unstable. There is a critical value of the forcing beyond all static pi-kinks fail to exist. Up to this value, the (in)stability of the pi-kinks can be established analytically in the strong coupling limits. Applying a forcing above the critical value causes the nucleation of 2pi-kinks and -antikinks. Besides a pi-kink, the unforced system also admits a static 3pi-kink. This state is unstable in the continuum models. By combining analytical and numerical methods in the discrete model, it is shown that the stable pi-kink remains stable, and that the unstable pi-kinks cannot be stabilized. The 3pi-kink does become stable in the discrete model when the coupling is sufficiently weak