Perturbation scheme for a fluxon in a curved Josephson junction

The kink solution in the long Josephson junction is studied. The perturbation scheme of constructing the fluxon solution in curved junction is formulated. The prediction from the perturbation scheme is compared with the prediction that follows from the numerical studies of the complete field model.

Figure 1

The curvature-dependent function $\mathcal{F}$ as a function of modulus of dimensionless quantity $aK$.

Figure 2

The first derivative of the field variable $\varphi$ for the kink solution in the flat junction (gray line) and in the curved junction (black line). The curvature in this plot is $aK=1.95$.

Figure 3

The static kink in the curved junction. The dashed gray line represents the curvature of the junction and the black continuous line represents the space gradient of the gauge-invariant phase difference. The configuration was stabilized before the instant $t=10$ and no change was observed until $t=100$. The parameters chosen in this plot are the following $A=1.9, \gamma =5$.

Figure 4

The space gradient of the gauge-invariant phase difference calculated from the perturbational scheme (gray line) and the numerical solution obtained in the model (31) (black line). The figures (a) $A=1$, (b) $A=1.5$, and (c) $A=1.9$ correspond to different values of the curvature parameter $A$.

Figure 5

The shape of the junction that corresponds to the curvature (30). The dashed line represents the curvature. The gray region represents the dielectric layer.

Figure 6

The final shape of the gradient of the gauge-invariant phase difference $\varphi$ at $t=1000$. The dashed line represents the curvature of the junction.

Figure 7

The evolution of the fluxon. (a) The fluxon leaves the potential hole at $t=10$. (b) Outside the curved region of the junction the fluxon became free $t=25$.