Exact analytical solution of the problem of current-carrying states of the Josephson junction in external magnetic fields

Abstract

The classical problem of the Josephson junction of arbitrary length W in the presence of externally applied magnetic fields (H) and transport currents (J) is reconsidered from the point of view of stability theory. In particular, we derive the complete infinite set of exact analytical solutions for the phase difference that describe the current-carrying states of the junction with arbitrary W and an arbitrary mode of the injection of J. These solutions are parameterized by two natural parameters: the constants of integration. The boundaries of their stability regions in the parametric plane are determined by a corresponding infinite set of exact functional equations. Being mapped to the physical plane (H, J), these boundaries yield the dependence of the critical transport current Jc on H. Contrary to a wide-spread belief, the exact analytical dependence Jc = Jc (H) proves to be multivalued even for arbitrarily small W . What is more, the exact solution reveals the existence of unquantized Josephson vortices carrying fractional flux and located near one of the junction edges, provided that J is sufficiently close to Jc for certain finite values of H. This conclusion (as well as other exact analytical results) is illustrated by a graphical analysis of typical cases.Solutions to (3), (4) are supposed to satisfy an obvious physical requirement: they must be stable with respect to any infinitesimal perturbations. (Unstable solutions that do not meet this requirement are physically unobservable and should be rejected.)Unfortunately, the standard boundary-value problem (3), (4) is mathematically ill-posed: 9 (i) for |J| larger than certain J max = J max (H, L), it does not admit any solutions at all; (ii) aside from stable (physical) solutions, there may exist unstable (unphysical) solutions for the same H and J; (iii) for the same H and J, there may exist several different physical solutions. An immediate consequence of this ill-posedness is as follows: although the general solution to (3) is well-known, 10 the constants of integration specifying particular physical solutions cannot be determined directly from the boundary conditions (4).In view of the above-mentioned mathematical difficulties, the previous analysis of the problem (3), (4) was concentrated mainly on finding the dependence J max = J max (H) (for particular values of L) without trying to establish the exact analytical form of current-carrying solutions. (It should be noted that the quantity J max itself was identified with the experimentally observable critical current J c , i.e., the identity J max ≡ J c was assumed.)For the case L ≪ 1, there existed 4 a simple analytical approximation for the dependence J max = J max (H) (the so-called 1,2,3 "Fraunhofer pattern"). As to the case L 1, only particular numerical results were obtained. Thus, Owen and Scalapino 6 established the dependence J max = J max (H) only for L = 5: it proved to be multivalued. The numerical method of Ref. 6 was later employed to study the effect of asymmetric injection of the transp...

The stationary sine-Gordon equation on metric graphs: Exact analytical solutions for simple topologies

“…(2)-(4) are consistent with other models of Josephson junction networks previously studied in [17,24,30,31]. Exact solutions of Eq.…”

“…Exact solutions of Eq. (1) on a finite interval have been obtained earlier in [17,30,31,34] for different special cases. Here, we use an approach similar to that of the Refs.…”

“…Here we briefly analyze the stability of the obtained solutions using the same method as in the Refs. [30,31]. We do this for a metric star graph, however, extending the method to tree graphs and other graph topologies is trivial.…”

Phase-coherent solitonic Josephson heat oscillator

“…Instead, for a magnetic field above the critical value, i.e., for H(t) > 2, solitons in the form of fluxons penetrate the LJJ from its ends. However, in this case at a specific value of the magnetic field several solutions, describing distinct configurations with different amount of solitons, may concurrently exist 51,[53][54][55] . The dynamical approach is essential to describe the JJ state when multiple solutions are available.…”

“…Nevertheless, for H(t) = 0 no solitons actually remain within the system in both forward and backward dynamics. This hysteretical behavior comes from the multistability of the SG model 33,51,[53][54][55][56] . In fact, Kuplevakhsky and Glukhov demonstrated that each solution of the SG equation, with a distinct number of solitons, is stable in a broad range of magnetic field values [53][54][55] .…”