Variational problems on multiply connected thin strips III: Integration of the Ginzburg-Landau equations over graphs

Rubinstein, Jacob; Schatzman, Michelle

doi:10.1090/S0002-9947-01-02804-5

Variational problems on multiply connected thin strips III: Integration of the Ginzburg-Landau equations over graphs
HTML articles powered by AMS MathViewer

by Jacob Rubinstein and Michelle Schatzman PDF

Trans. Amer. Math. Soc. 353 (2001), 4173-4187 Request permission

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 Ginzburg-Landau 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.

References

H.J. Carlin and A.B. Giordano. Network Theory. Prentice Hall, 1964.
Takayuki Furuta, Parametric operator function via Furuta inequality, Sci. Math. 1 (1998), no. 1, 1–5. MR 1688425
W.A. Little and R.D. Parks. Observation of quantum periodicity in the transition temperature of a superconducting cylinder. Phys. Rev. Lett., 9:9–12, 1962.
V.V. Moshchalkov, L.Gielen, and Y.Bruynseraede. Effect of sample topology on the critical fields of mesoscopic superconductors. Nature, 373(6512):319, 1995.
B.Pannetier. Superconducting wire networks. In B.Kramer, editor, Quantum Coherence in Mesoscopic Systems, pages 457–484. Plenum Press, 1991.
J.Rubinstein and M.Schatzman. Variational problems on multiply connected thin strips IV: Zero sets for the Ginzburg-Landau linearized equations. in preparation.
J.Rubinstein and M.Schatzman. Variational problems on multiply connected thin strips II: convergence of the Ginzburg - Landau functional. Technical Report 294, UMR 5585 CNRS Equipe d’Analyse Numérique, April 1999.