  Spin Hall oscillators (SHO) are promising candidates for the generation,
detection and amplification of high frequency signals, that are tunable through
a wide range of operating frequencies. They offer to be read out electrically,
magnetically and optically in combination with a simple bilayer design. Here,
we experimentally study the spatial dependence and spectral properties of
auto-oscillations in SHO devices based on Pt(7 nm)/\
Ni$_{\mathrm{80}}$Fe$_{\mathrm{20}}$(5 nm) tapered nanowires. Using Brillouin
light scattering microscopy, we observe two individual self-localized spin-wave
bullets that oscillate at two distinct frequencies (5.2 GHz and 5.45 GHz) and
are localized at different positions separated by about 750 nm within the SHO.
This state of a tapered SHO has been predicted by a Ginzburg-Landau
auto-oscillator model, but not yet been directly confirmed experimentally. We
demonstrate that the observed bullets can be individually synchronized to
external microwave signals, leading to a frequency entrainment, linewidth
reduction and increase in oscillation amplitude for the bullet that is selected
by the microwave frequency. At the same time, the amplitude of other parasitic
modes decreases, which promotes the single-mode operation of the SHO. Finally,
the synchronization of the spin-wave bullets is studied as a function of the
microwave power. We believe that our findings promote the realization of
extended spin Hall oscillators accomodating several distinct spin-wave bullets,
that jointly cover an extended range of tunability.
