  We show that the solutions of the Bogomolny equations for the Abelian Higgs
model on a two-dimensional torus, can be expanded in powers of a quantity
epsilon measuring the departure of the area from the critical area. This allows
a precise determination of the shape of the solutions for all magnetic fluxes
and arbitrary position of the Higgs field zeroes. The expansion is carried out
to 51 orders for a couple of representative cases, including the unit flux
case. We analyse the behaviour of the expansion in the limit of large areas, in
which case the solutions approach those on the plane. Our results suggest
convergence all the way up to infinite area.
