Asymptotic analysis of the one-dimensional Ginzburg-Landau equations near self-duality

Almog, Y.

doi:10.1090/qam/1686194

Author: Y. Almog
Journal: Quart. Appl. Math. 57 (1999), 355-367
MSC: Primary 35Q55; Secondary 34B15, 34E20, 82D55
DOI: https://doi.org/10.1090/qam/1686194
MathSciNet review: MR1686194
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Abstract: It is known that when the Ginzburg-Landau parameter $k = 1/\sqrt 2$ the one-dimensional Ginzburg-Landau equations exhibit self-duality and may be reduced into a pair of first-order ODE. The present asymptotic analysis initially focuses on infinite samples of superconductors for which $\left | {k - 1/\sqrt 2 } \right | \ll 1$. It is shown that when the value of the applied magnetic field at infinity lies between $k$ and $1/\sqrt 2$ a superconducting solution exists. It is later shown, for arbitrary values of $k$, that no solution, other than the normal state, can exist for applied magnetic field values that lie outside the above interval.

Mel S. Berger, Creation and breaking of self-duality symmetry—a modern aspect of calculus of variations, Nonlinear partial differential equations (Durham, N.H., 1982) Contemp. Math., vol. 17, Amer. Math. Soc., Providence, RI, 1983, pp. 379–394. MR 706111, DOI https://doi.org/10.1090/conm/017/706111
E. B. Bogomol′nyĭ, The stability of classical solutions, Jadernaja Fiz. 24 (1976), no. 4, 861–870 (Russian); English transl., Soviet J. Nuclear Phys. 24 (1976), no. 4, 449–454. MR 0443719
C. Bolley and B. Helffer, Rigorous results on Ginzburg-Landau models in a film submitted to an exterior parallel magnetic field. I, Nonlinear Stud. 3 (1996), no. 1, 1–29. MR 1396033
C. Bolley and B. Helffer, Rigorous results on Ginzburg-Landau models in a film submitted to an exterior parallel magnetic field. II, Nonlinear Stud. 3 (1996), no. 2, 121–152. MR 1422817
S. J. Chapman, Nucleation of superconductivity in decreasing fields. I, II, European J. Appl. Math. 5 (1994), no. 4, 449–468, 469–494. MR 1309734, DOI https://doi.org/10.1017/s0956792500001571
S. J. Chapman, Nucleation of superconductivity in decreasing fields. I, II, European J. Appl. Math. 5 (1994), no. 4, 449–468, 469–494. MR 1309734, DOI https://doi.org/10.1017/s0956792500001571
S. J. Chapman, Asymptotic analysis of the Ginzburg-Landau model of superconductivity: reduction to a free boundary model, Quart. Appl. Math. 53 (1995), no. 4, 601–627. MR 1359498, DOI https://doi.org/10.1090/qam/1359498
S. J. Chapman, S. D. Howison, J. B. McLeod, and J. R. Ockendon, Normal/superconducting transitions in Landau-Ginzburg theory, Proc. Roy. Soc. Edinburgh Sect. A 119 (1991), no. 1-2, 117–124. MR 1130600, DOI https://doi.org/10.1017/S0308210500028353

A. T. Dorsey, Dynamics of interfaces in superconductors, Ann. Phys. 233, 248–269 (1994)

V. A. Ginzburg and L. D. Landau, On the theory of superconductivity, Soviet Phys. JETP 20, 1064–1082 (1950)

J. Kevorkian and Julian D. Cole, Perturbation methods in applied mathematics, Applied Mathematical Sciences, vol. 34, Springer-Verlag, New York-Berlin, 1981. MR 608029
Martin H. Millman and Joseph B. Keller, Perturbation theory of nonlinear boundary-value problems, J. Mathematical Phys. 10 (1969), 342–361. MR 237867, DOI https://doi.org/10.1063/1.1664849
J. D. Murray, Asymptotic analysis, 2nd ed., Applied Mathematical Sciences, vol. 48, Springer-Verlag, New York, 1984. MR 740864