# Stability analysis of π-kinks in a 0-π Josephson junction

G. Derks, A. Doelman, Stephanus A. van Gils, H. Susanto

11 Citations (Scopus)

### 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.
Original language Undefined 10.1137/060657984 99-141 43 SIAM journal on applied dynamical systems 6 1 https://doi.org/10.1137/060657984 Published - 2007

• METIS-247074
• MSC-34D35
• EWI-11491
• MSC-35Q53
• MSC-37K50
• MSC-39A11
• IR-62039

### Cite this

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title = "Stability analysis of π-kinks in a 0-π Josephson junction",
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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author = "G. Derks and A. Doelman and {van Gils}, {Stephanus A.} and H. Susanto",
note = "10.1137/060657984",
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Stability analysis of π-kinks in a 0-π Josephson junction. / Derks, G.; Doelman, A.; van Gils, Stephanus A.; Susanto, H.

In: SIAM journal on applied dynamical systems, Vol. 6, No. 1, 10.1137/060657984, 2007, p. 99-141.

TY - JOUR

T1 - Stability analysis of π-kinks in a 0-π Josephson junction

AU - Derks, G.

AU - Doelman, A.

AU - van Gils, Stephanus A.

AU - Susanto, H.

N1 - 10.1137/060657984

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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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KW - MSC-35Q53

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