Asymptotic analysis of the one-dimensional Ginzburg-Landau equations near self-duality
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.
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M. S. Berger, Creation and breaking of self-duality symmetry—a modern aspect of calculus of variations, Contemp. Math. 17, Amer. Math. Soc., Providence, RI, 1983, pp. 379–394
E. Bogomolnyi, The stability of classical solutions, Sov. Jour. Nucl. Phys. 24, 449–454 (1976)
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, 1–29 (1996)
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, 121–152 (1996)
S. J. Chapman, Nucleation of superconductivity in decreasing fields I, EJAM 5, 449–468 (1994)
S. J. Chapman, Nucleation of superconductivity in decreasing fields II, EJAM 5, 469–494 (1994)
S. J. Chapman, Asymptotic analysis of the Ginzburg-Landau model of superconductivity: reduction to a free boundary model, Quart. Appl. Math. 53, 601–627 (1995)
S. J. Chapman, S. D. Howison, J. B. Mcleod, and J. R. Ockendon, Normal/Superconducting transition in Landau-Ginzburg Theory, Proc. Roy. Soc. Edinburgh 119A, 117–124 (1991)
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 J. D. Cole, Perturbation Methods in Applied Mathematics, Applied Mathematical Sciences, vol. 34, Springer-Verlag, New York, 1981
M. H. Millman and J. B. Keller, Perturbation theory of nonlinear boundary value problem, J. Math. Phys. 10, 342–361 (1969)
J. D. Murray, Asymptotic Analysis, second edition, Applied Mathematical Sciences, vol. 48, Springer-Verlag, New York, 1984
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© Copyright 1999
American Mathematical Society