We analyze the motion of quantum vortices in a two-dimensional spinless superfluid within Popovs hydrodynamic description. In the long healing length limit (where a large number of particles are inside the vortex core) the superfluid dynamics is dete
rmined by saddle points of Popovs 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.
