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

    13 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

    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.

    KW - METIS-247074

    KW - MSC-34D35

    KW - EWI-11491

    KW - MSC-35Q53

    KW - MSC-37K50

    KW - MSC-39A11

    KW - IR-62039

    U2 - 10.1137/060657984

    DO - 10.1137/060657984

    M3 - Article

    VL - 6

    SP - 99

    EP - 141

    JO - SIAM journal on applied dynamical systems

    JF - SIAM journal on applied dynamical systems

    SN - 1536-0040

    IS - 1

    M1 - 10.1137/060657984

    ER -