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Journal of Vibration Testing and System Dynamics 8(3) (2024) 317--328 | DOI:10.5890/JVTSD.2024.09.004

Mei-Qin Zhan, Kening Wang

Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, USA

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Abstract

It is well-known that the Ginzburg-Landau equations admit at least three time-periodic solutions. One of them describes the non-super- conductive (or normal) state and the other one describes the superconductivity state. In this paper, we investigate the uniform boundedness and attractivity of these time-periodic solutions. Moreover, numerical approximations to time-periodic solutions are also presented.

References

1. [1]	Wang, S. and Zhan, M. (1999), $L^p$ Solutions to time dependent Ginzburg-Landau equations of superconductivity, Nonlinear Analysis: Theory, Methods $\&$ Applications, 36, 661-677

2. [2]	de Gennes, P. (1966), Superconductivity in Metals and Alloys, Benjamin, New York.

3. [3]	Beasley, M.R. (2009) Notes on the Ginzburg-Landau Theory, ICMR Summer School on Novel SuperconductorsUniversity of California, Santa Barbara.

4. [4]	Konsin, P. and Sorkin, B. (2009) Time-dependent Ginzburg-Landau equations for a two-component superconductor and the doping dependence of the relaxation times of the order parameters in $YBa_2Cu_3O_{7-\delta}$, Journal of Physics: Conference Series, 150.

5. [5]	Fan, J., Nakamura, G., and Zhan, M. (2021), Uniqueness of weak solutions to phase-lock equations, Discontinuity, Nonlinearity, and Complexity, 10(1), 31-41.

6. [6]	Wang, B. (1999), Existence of Time Periodic Solutions for the Ginzburg-Landau equations of superconductivity, Journal of Mathematical Analysis and Applications, 232(2), 394-412

7. [7]	Zhan, M. (2000), Existence of periodic solutions for Ginzburg-Landau equations of superconductivity, Journal of Mathematical Analysis and Applications, 249(2), 614-625

8. [8]	Zhan, M. (2008), Multiplicity and stability of time-periodic solutions of Ginzburg-Landau equations of superconductivity, Journal of Mathematical Analysis and Applications, 340, 126–134

9. [9]	Temam, R. (2000), Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag.

10. [10]	Pao, C.V. (1999), Periodic solutions of parabolic systems with nonlinear boundary conditions, Journal of Mathematical Analysis and Applications, 234, 695-716

11. [11]	Rahman, M., Wang, K., and Zhan, M. Time-periodic Solutions of Ordinary Differential Equations of Superconductivity, preprint.