Split Mode Method for the Elliptic 2D Sine-Gordon Equation: Application to Josephson Junction in Overlap Geometry

Jean-Guy Caputo, Nikos Flytzanis, Yuri Gaididei, Irene Moulitsa and Emmanuel Vavalis
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Jean-Guy Caputo: Laboratoire de Mathématiques, Institut de Sciences Appliquees and U.P.R.E.S.A. C.N.R.S. 60–85, BP8, 76131 Mont-Saint-Aignan Cedex, France
Nikos Flytzanis: Physics Department, University of Crete, 71409 Heraklion, Greece
Yuri Gaididei: Institute for Theoretical Physics, 252143 Kiev, Ukraine
Irene Moulitsa: Mathematics Department, University of Crete, 71409 Heraklion, Greece;
Emmanuel Vavalis: Mathematics Department, University of Crete, 71409 Heraklion, Greece;

International Journal of Modern Physics C (IJMPC), 1998, vol. 09, issue 02, 301-323

Abstract: We introduce a new type of splitting method for semilinear partial differential equations. The method is analyzed in detail for the case of the two-dimensional static sine-Gordon equation describing a large area Josephson junction with overlap current feed and external magnetic field. The solution is separated into an explicit term that satisfies the one-dimensional sine-Gordon equation in they-direction with boundary conditions determined by the bias current and a residual which is expanded using modes in they-direction, the coefficients of which satisfy ordinary differential equations inxwith boundary conditions given by the magnetic field. We show by direct comparison with a two-dimensional solution that this method converges and that it is an efficient way of solving the problem. The convergence of theyexpansion for the residual is compared for Fourier cosine modes and the normal modes associated to the static one-dimensional sine-Gordon equation and we find a faster convergence for the latter. Even for such large widths asw=10two such modes are enough to give accurate results.

Keywords: Josephson Junctions; Partial Differential Equations; Solitons; 74.50.+r; 02.60.Lj; 02.70.Dh (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1142/S0129183198000236

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