Phase-slip lines can be viewed as dynamically created Josephson junctions in a homogeneous superconducting film. In contrast to phase-slip centers, phase-slip lines occur in wide superconducting strips, where the order parameter may vary in two dimensions. We investigated phase-slip lines in two different materials using several methods. We observed Shapiro steps under microwave radiation, which shows that the frequency of the order parameter oscillation is equal to Josephson frequency. A periodic oscillation of a critical current versus the applied magnetic field was found in strips with a hole in the middle. The latter effect provides a clear evidence of macroscopic quantum interference across a phase-slip line. We have used low temperature scanning laser microscopy to visualize the phase-slip lines and to distinguish them from possible local inhomogeneities in the films.

The problem of vortex structure in a single Josephson junction in an external magnetic field, in the absence of transport currents, is reconsidered from a new mathematical point of view. In particular, we derive a complete set of exact analytical solutions representing all the stationary points (minima and saddle-points) of the relevant Gibbs free-energy functional. The type of these solutions is determined by explicit evaluation of the second variation of the Gibbs free-energy functional. The stable (physical) solutions minimizing the Gibbs free-energy functional form an infinite set and are labelled by a topological number Nv = 0, 1, 2, . . .. Mathematically, they can be interpreted as nontrivial "vacuum" (Nv = 0) and static topological solitons (Nv = 1, 2, . . .) of the sine-Gordon equation for the phase difference in a finite spatial interval: solutions of this kind were not considered in previous literature. Physically, they represent the Meissner state (Nv = 0) and Josephson vortices (Nv = 1, 2, . . .). Major properties of the new physical solutions are thoroughly discussed. An exact, closed-form analytical expression for the Gibbs free energy is derived and analyzed numerically. Unstable (saddle-point) solutions are also classified and discussed.

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...