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

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

Research output: Contribution to journalArticleAcademicpeer-review

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 languageUndefined
Article number10.1137/060657984
Pages (from-to)99-141
Number of pages43
JournalSIAM journal on applied dynamical systems
Volume6
Issue number1
DOIs
Publication statusPublished - 2007

Keywords

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

Cite this

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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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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.

Research output: Contribution to journalArticleAcademicpeer-review

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

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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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