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


Derks, G. and Doelman, A. and Gils, S.A. van and Susanto, H. (2007) Stability analysis of $\pi$-kinks in a 0-$\pi$ Josephson junction. SIAM Journal on Applied Dynamical Systems (SIADS), 6 (1). pp. 99-141. ISSN 1536-0040

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Abstract:We consider a spatially nonautonomous 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 nonautonomous character is due to the presence of a discontinuity point, namely, a jump of $\pi$ in the sine-Gordon phase. The continuum model 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 kinks is stable and the others are unstable. There is a critical value of the forcing beyond which 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 2$\pi$-kinks and -antikinks. Besides a $\pi$-kink, the unforced system also admits a static 3$\pi$-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 by decreasing the coupling. The 3$\pi$-kink does become stable in the discrete model when the coupling is sufficiently weak.
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Electrical Engineering, Mathematics and Computer Science (EEMCS)
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