  We investigate analytically and numerically the stability of bubble-like
fluxons in disk-shaped heterogeneous Josephson junctions. Using ring solitons
as a model of bubble fluxons in the two-dimensional sine-Gordon equation, we
show that the insertion of coaxial dipole currents prevents their collapse. We
characterize the onset of instability by introducing a single parameter that
couples the radius of the bubble fluxon with the properties of the injected
current. For different combination of parameters, we report the formation of
stable oscillating bubbles, the emergence of internal modes, and bubble breakup
due to internal mode instability. We show that the critical germ depends on the
ratio between its radius and the steepness of the wall separating the different
phases in the system. If the steepness of the wall is increased (decreased),
the critical radius decreases (increases). Our theoretical findings are in good
agreement with numerical simulations. We discuss applications in quantum
information technologies.
