0- $π$约瑟夫森结中$π$ -扭结的稳定性分析
Stability analysis of $π$-kinks in a 0-$π$ Josephson junction
摘要：
我们考虑了一个空间非自治的离散sine-Gordon方程，该方程具有常数强迫和连续极限，用来模拟具有外加偏置电流的0- $π$约瑟夫森结。连续极限对应于离散系统的强耦合极限。非自治的字符是由于一个不连续点的存在，即$π$在sine-Gordon相位的跳跃。连续介质模型允许静态孤立波，称之为$π$ -扭结，并附加到不连续点。对于小的强迫，有三种类型的$π$ -扭结。我们表明，扭结之一是稳定的，其他的是不稳定的。在所有静态$π$之外存在一个临界强制值-扭结不存在。直到这个值，$π$ -扭结的（不）稳定性可以在强耦合极限中解析地建立。施加高于临界值的力导致$2π$ -扭结和-反扭结的成核。除了$π$扭折外，非受迫系统还存在静态$3π$扭折.这种状态在连续介质模型中是不稳定的。通过结合分析和数值方法的离散模型，它表明，稳定的$π$ -扭结保持稳定，而不稳定的$π$ -扭结不能通过减少耦合稳定。当耦合足够弱时，离散模型中的$3π$扭结确实变得稳定。
英文摘要：
We consider a spatially non-autonomous discrete sine-Gordon equation with constant forcing and its continuum limit(s) to model a 0-$π$ 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 $π$ in the sine-Gordon phase. The continuum models admits static solitary waves which are called $π$-kinks and are attached to the discontinuity point. For small forcing, there are three types of $π$-kinks. We show that one of the kinks is stable and the others are unstable. There is a critical value of the forcing beyond all static $π$-kinks fail to exist. Up to this value, the (in)stability of the $π$-kinks can be established analytically in the strong coupling limits. Applying a forcing above the critical value causes the nucleation of $2π$-kinks and -antikinks. Besides a $π$-kink, the unforced system also admits a static $3π$-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 $π$-kink remains stable, and that the unstable $π$-kinks cannot be stabilized by decreasing the coupling. The $3π$-kink does become stable in the discrete model when the coupling is sufficiently weak.
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