  A model derived in [14] for n near-parallel vortex filaments in a three
dimensional fluid region takes into consideration the pairwise interaction
between the filaments along with an approximation for motion by self-induction.
The same system of equations appears in descriptions of the fine structure of
vortex filaments in the Gross -- Pitaevski model of Bose -- Einstein
condensates. In this paper we construct families of standing waves for this
model, in the form of n co-rotating near-parallel vortex filaments that are
situated in a central configuration. This result applies to any pair of vortex
filaments with the same circulation, corresponding to the case n=2. The model
equations can be formulated as a system of Hamiltonian PDEs, and the
construction of standing waves is a small divisor problem. The methods are a
combination of the analysis of infinite dimensional Hamiltonian dynamical
systems and linear theory related to Anderson localization. The main technique
of the construction is the Nash-Moser method applied to a Lyapunov-Schmidt
reduction, giving rise to a bifurcation equation over a Cantor set of
parameters.
