Bifurcations of the solutions of modified Ginzburg–Landau equation for Josephson junctions

In this paper we investigate numerically a class of Josephson contacts with plane boundaries on the basis of modified Ginzburg–Landau (GL) type equations. The corresponding non-linear boundary value problem (BVP) for the amplitude of the order parameter is solved numerically using the continuous analog of Newton method coupled with finite element method. We show, that for fixed values of the phenomenological coefficients of the contact there exist various solutions with different energies and their own phase differences. The resulted supercurrent density – phase offset curves are constructed numerically for different phenomenological coefficients. We show, that each curve consists of three smoothly joined branches corresponding to stable and unstable states of the order parameter's amplitude. The critical Josephson (super) current appears to be a bifurcation value for these states and conforms to the points of confluence of the separated branches. In order to estimate the influence of the phenomenological coefficients on the form of such curves, a Fourier decomposition is made. We show, that due to existence of different nonlinear terms in the equation described the amplitude of the order parameter in $S$ and $N$ regions, the supercurrent density – phase offset dependence is sinusoidal only for restricted domain of values of phenomenological coefficients. In particular, in the case of large difference between the effective masses in $S$ and $N$ regions the availability of the second harmonic is substantial.