SPLIT MODE METHOD FOR THE ELLIPTIC 2D SINE-GORDON EQUATION - APPLICATION TO JOSEPHSON-JUNCTION IN OVERLAP GEOMETRY

We introduce a new type of splitting method for semilinear partial dif ferential equations. The method is analyzed in detail for the case of the two-dimensional static sine-Gordon equation describing a large are a Josephson junction with overlap current feed and external magnetic f ield. The solution is separated into an explicit term that satisfies t he one-dimensional sine-Gordon equation in the y-direction with bounda ry conditions determined by the bias current and a residual which is e xpanded using modes in the y-direction, the coefficients of which sati sfy ordinary differential equations in x with boundary conditions give n by the magnetic field. We show by direct comparison with a two-dimen sional solution that this method converges and that it is an efficient way of solving the problem. The convergence of the y expansion for th e residual is compared for Fourier cosine modes and the normal modes a ssociated to the static one-dimensional sine-Gordon equation and we fi nd a faster convergence for the latter. Even for such large widths as w = 10 two such modes are enough to give accurate results.