Numerical investigation of a bifurcation problem with free boundaries arising from the physics of josephson junctions

A direct method for calculating the minimal length of “one-dimensionaf” long homogeneous or inhomogeneous Josephson junction in which the specific distribution of the magnetic flux retains its stability is proposed. Since the length of the junction is a variable quantity, the corresponding nonlinear spectral problem as a problem with free boundaries is interpreted.
The obtained results give us warranty to consider as “long”, every Josephson junction in which there exists at least one nontrivial stable distribution of the magnetic flux. If the junction is inhomogeneous there is an optimal width of the inhomogeinity for which the minimal junction length providing a stable soliton becomes minimal for fixed values of the all other parameters.