  Temporal solitons are optical pulses that arise from the balance of negative
group-velocity dispersion and self-phase modulation. For decades only quadratic
dispersion was considered, with higher order dispersion thought of as a
nuisance. Following the recent reporting of pure-quartic solitons, we here
provide experimental and numerical evidence for an infinite hierarchy of
solitons that balance self-phase modulation and arbitrary negative pure,
even-order dispersion. Specifically, we experimentally demonstrate the
existence of solitons with pure-sextic ($\beta_6$), -octic ($\beta_8$) and
-decic ($\beta_{10}$) dispersion, limited only by the performance of our
components, and show numerical evidence for the existence of solitons involving
pure $16^{\rm th}$ order dispersion. Phase-resolved temporal and spectral
characterization reveals that these pulses, exhibit increasing spectral
flatness with dispersion order. The measured energy-width scaling laws suggest
dramatic advantages for ultrashort pulses. These results broaden the
fundamental understanding of solitons and present new avenues to engineer
ultrafast pulses in nonlinear optics and its applications.
