Pinned fluxons in a Josephson junction with a finite-length inhomogeneity

Abstract

We consider a Josephson junction system installed with a finite length inhomogeneity, either of microresistor or of microresonator type. The system can be modelled by a sine-Gordon equation with a piecewise-constant function to represent the varying Josephson tunneling critical current. The existence of pinned fluxons depends on the length of the inhomogeneity, the variation in the Josephson tunneling critical current and the applied bias current. We establish that a system may either not be able to sustain a pinned fluxon, or -for instance by varying the length of the inhomogeneity -may exhibit various different types of pinned fluxons. Our stability analysis shows that changes of stability can only occur at critical points of the length of the inhomogeneity as a function of the (Hamiltonian) energy density inside the inhomogeneity -a relation we determine explicitly. In combination with continuation arguments and Sturm-Liouville theory, we determine the stability of all constructed pinned fluxons. It follows that if a given system is able to sustain at least one pinned fluxon, there is exactly one stable pinned fluxon, i.e. the system selects one unique stable pinned configuration. Moreover, it is shown that both for microresistors and microresonators this stable pinned configuration may be non-monotonic -something which is not possible in the homogeneous case. Finally, it is shown that results in the literature on localised inhomogeneities can be recovered as limits of our results on microresonators.

A Geometric Approach to Stationary Defect Solutions in One Space Dimension

“…Besides pinning a solution, a defect may also annihilate, rebound, penetrate, or split a (traveling) solution, and it may correspond to a source (or sink) sending out (or absorbing) traveling waves; see, for example, [15,16,31,42,64]. We do not consider these phenomena in the current manuscript.…”

“…However, one may also expect pinned stationary defect solutions in heterogeneous PDEs for which the homogeneous limit does not have a corresponding stationary localized pattern. Typically, there does exist a traveling localized structure in the homogeneous limit in such cases, which indeed gets pinned by the defect and thus corresponds to the stationary solution of the heterogeneous system; see, for example, [15,16,31,41].…”

Stability of Fronts in Inhomogeneous Wave Equations

“…More examples with N = 1 and N = 2 can be found in [6,22]. This example is similar to the example in [23] and can be related to long Josephson junctions with defects.…”

“…Another approximation for this physical reality are inhomogeneities of a tiny length L i and potentials of the form V i (u) = u/L i . In [6], it is shown that the stationary fronts associated with localised homogeneities as described in McLaughlin & Scott [27] can be embedded in the family of stationary waves of finite length inhomogeneities. The stability properties follow from the theory presented in this paper and they regain the criterion derived in [27] (as well as the stability of finite length inhomogeneities).…”

“…We define (u 1 , p 1 ) to be the matching point between I ℓ and I m1 ; (u 2 , p 2 ) the matching point between I m1 and I m2 ; and (u 3 , p 3 ) the matching point between I m2 and I r . The Hamiltonian formulation (6) gives that these points can be expressed as functions of the Hamiltonian parameters h i :…”