Stability analysis of static solutions in a Josephson junction

We present all the possible solutions of a Josephson junction with bias current and magnetic field with both inline and overlap geometry, and examine their stability. We follow the bifurcation of new solutions as we increase the junction length. The analytical results are in terms of elliptic functions for the case of inline geometry, and are in agreement with the numerical calculations, explaining also the strong hysteretic phenomena typically seen in the calculation of the maximum tunnelling current. This suggests a different experimental approach based on the use, instead of the external magnetic field, of the modulus of the elliptic function or the related quantity the total magnetic flux to avoid hysteretic behaviour and unfold the overlapping Imax (H ) curves.