  We consider the propagation of breathers along a quantised superfluid vortex.
Using the correspondence between the local induction approximation (LIA) and
the nonlinear Schr\"odinger equation, we identify a set of initial conditions
corresponding to breather solutions of vortex motion governed by the LIA. These
initial conditions, which give rise to a long-wavelength modulational
instability, result in the emergence of large amplitude perturbations that are
localised in both space and time. The emergent structures on the vortex
filament are analogous to loop solitons. Although the breather solutions we
study are exact solutions of the LIA equations, we demonstrate through full
numerical simulations that their key emergent attributes carry over to vortex
dynamics governed by the Biot-Savart law and to quantized vortices described by
the Gross-Pitaevskii equation. The breather excitations can lead to
self-reconnections, a mechanism that can play an important role within the
cross-over range of scales in superfluid turbulence. Moreover, the observation
of breather solutions on vortices in a field model suggests that these
solutions are expected to arise in a wide range of other physical contexts from
classical vortices to cosmological strings.
