  A numerical simulation is carried out for static vortices in a long Josephson
junction with an exponentially varying width. At specified values of the
parameters the corresponding boundary-value problem admits more than one
solution. Each solution (distribution of the magnetic flux in the junction) is
associated to a Sturm--Liouville problem, the smallest eigenvalue of which can
be used, in a first approximation, to assess the stability of the vortex
against relatively small spatiotemporal perturbations. The change in width of
the junction leads to a renormalization of the magnetic flux in comparison with
the case of a linear one-dimensional model. The influence of the model
parameters on the stability of the states of the magnetic flux is investigated
in detail, particularly that of the shape parameter. The critical curve of the
junction is constructed from pieces of the critical curves for the different
magnetic flux distributions having the highest critical currents for the given
magnetic field.

Superconductivity