Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A modified Ginzburg-Landau model for Josephson junctions in a ring

Authors: E. Hill, J. Rubinstein and P. Sternberg
Journal: Quart. Appl. Math. 60 (2002), 485-503
MSC: Primary 35Q60; Secondary 35B27, 35J20, 82D55
DOI: https://doi.org/10.1090/qam/1914438
MathSciNet review: MR1914438
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Ginzburg-Landau functional to analyze SNS junctions in a one-dimensional ring. We compare several canonical scalings. The linearized problem is solved to obtain the phase transition curves. We compute the $ \Gamma $-limit of the functional in the different scalings. The interaction of several junctions is analyzed. We study the zero set of the order parameter for distinguished values of the flux. Finally, we compute the currents in the weakly nonlinear regime.

References [Enhancements On Off] (What's this?)

  • [1] H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, MA, 1984 MR 773850
  • [2] A. Baratoff, J. A. Blackburn and B. B. Schwarz, Current-phase relationship in short superconducting weak links, Phys. Rev. Lett. 16, 1096-1099 (1970)
  • [3] J. Berger and J. Rubinstein, Topology of the order parameter in the Little Parks experiment, Phys. Rev. Lett. 75, 320-322 (1995)
  • [4] J. Berger and J. Rubinstein, Bifurcation analysis for phase transitions in nonuniform superconducting rings, SIAM J. Appl. Math. 58, 103-121 (1998) MR 1610025
  • [5] J. Berger and J. Rubinstein, The zero set of the order parameter wave function, Comm. Math. Phys. 202, 621-628 (1999) MR 1690956
  • [6] S. J. Chapman, Nucleation of superconductivity in decreasing fields, I, European J. Appl. Math. 5, 449-468 (1994) MR 1309734
  • [7] S. J. Chapman, Q. Du and M. D. Gunzburger, A Ginzburg Landau model of superconducting / normal junctions including Josephson junctions, European J. Appl. Math. 6, 97-114 (1996) MR 1331493
  • [8] P. G. de Gennes, Superconductivity of Metals and Alloys, Addison-Wesley, 1989
  • [9] Q. Du and J. Remski, Simplified models for superconducting-normal-superconducting junctions and their numerical approximations, European J. Appl. Math. 10, 1-25 (1999) MR 1685818
  • [10] B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and M.P. Owen, Nodel sets for ground-states of Schrödinger operators with zero magnetic field in non-simply connected domains, Comm. Math. Phys. 202, 629-649 (1999) MR 1690957
  • [11] B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and M. P. Owen, Nodal sets, multiplicity and superconductivity in non-simply connected domains, in Connectivity and superconductivity, eds. J. Berger and J. Rubinstein, Springer Lecture Notes in Physics, vol. M62, 2000
  • [12] E. Hill, A Ginzburg-Landau model for Josephson junctions in a ring, Ph.D. thesis, Indiana University, July, 2001 MR 2702461
  • [13] K. H. Hoffman, L. Jiang and W. Yu, Models of superconducting-normal-superconducting junctions, Math. Methods. Appl. Sci. 21, 59-91 (1998) MR 1487827
  • [14] K. Likharev, Superconducting weak links, Rev. Mod. Phys. 51, 101-159 (1979)
  • [15] 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)
  • [16] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, Inc., 1978 MR 0493421
  • [17] J. Rubinstein and M. Schatzman, Variational problems in multiply connected thin strips II: The asymptotic limit of the Ginzburg-Landau functional, Arch. Rational Mech. Anal., to appear.
  • [18] J. Rubinstein and P. Sternberg, in preparation.
  • [19] M. Tinkham, Introduction to Superconductivity, McGraw Hill, New York, 1996

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35Q60, 35B27, 35J20, 82D55

Retrieve articles in all journals with MSC: 35Q60, 35B27, 35J20, 82D55

Additional Information

DOI: https://doi.org/10.1090/qam/1914438
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society