  We revisit the problem of transverse instability of a 2D breather stripe of
the sine-Gordon (sG) equation. A numerically computed Floquet spectrum of the
stripe is compared to analytical predictions developed by means of
multiple-scale perturbation theory showing good agreement in the
long-wavelength limit. By means of direct simulations, it is found that the
instability leads to a breakup of the quasi-1D breather in a chain of
interacting 2D radial breathers that appear to be fairly robust in the
dynamics. The stability and dynamics of radial breathers in a finite domain are
studied in detail by means of numerical methods. Different families of such
solutions are identified. They develop small-amplitude spatially oscillating
tails ("nanoptera") through a resonance of higher-order breather's harmonics
with linear modes ("phonons") belonging to the continuous spectrum. These
results demonstrate the ability of the 2D sG model within our finite domain
computations to localize energy in long-lived, self-trapped breathing
excitations.
