  We analyze the motion of quantum vortices in a two-dimensional spinless
superfluid within Popov's hydrodynamic description. In the long healing length
limit (where a large number of particles are inside the vortex core) the
superfluid dynamics is determined by saddle points of Popov's action, which, in
particular, allows for weak solutions of the Gross-Pitaevskii equation. We
solve the resulting equations of motion for a vortex moving with respect to the
superfluid and find the reconstruction of the vortex core to be a non-analytic
function of the force applied on the vortex. This response produces an
anomalously large dipole moment of the vortex and, as a result, the spectrum
associated with the vortex motion exhibits narrow resonances lying {\em within}
the phonon part of the spectrum, contrary to traditional view.
