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We provide here new insights into the classical problem of a one-dimensional superconducting wire exposed to an applied electric current using the time-dependent Ginzburg-Landau model. The most striking feature of this system is the well-known appearance of oscillatory solutions exhibiting phase slip centers (PSC's) where the order parameter vanishes. Retaining temperature and applied current as parameters, we present a simple yet definitive explanation of the mechanism within this nonlinear model that leads to the PSC phenomenon and we establish where in parameter space these oscillatory solutions can be found. One of the most interesting features of the analysis is the evident collision of real eigenvalues of the associated PT-symmetric linearization, leading as it does to the emergence of complex elements of the spectrum.

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Abstract. We analyze the one-dimensional Ginzburg-Landau functional of superconductivity on a planar graph. In the Euler-Lagrange equations, the equation for the phase can be integrated, provided that the order parameter does not vanish at the vertices; in this case, the minimization of the GinzburgLandau functional is equivalent to the minimization of another functional, whose unknowns are a real-valued function on the graph and a finite set of integers.