  The frustrated rare-earth pyrochlore Yb$_2$Ti$_2$O$_7$ is remarkable among
magnetic materials: despite a ferromagnetically ordered ground state it
exhibits a broad, nearly gapless, continuum of excitations. This broad
continuum connects smoothly to the sharp one-magnon excitations expected, and
indeed observed, at high magnetic fields, raising the question: how does this
picture of sharp magnons break down as the field is lowered? In this paper, we
consider the effects of magnon interactions in Yb$_2$Ti$_2$O$_7$, showing that
their inclusion greatly extends the reach of spin-wave theory. First, we show
that magnon interactions shift the phase boundary between the (splayed)
ferromagnet (SFM) and the antiferromagnetic $\Gamma_5$ phase so that
Yb$_2$Ti$_2$O$_7$ lies very close to it. Next, we show how the high-field limit
connects to lower fields; this includes corrections to the critical fields for
the $[111]$ and $[1\bar{1}0]$ directions, bringing them closer to the observed
experimental values, as well as accounting for the departures from linear spin
wave theory that appear in $[001]$ applied fields below 3 T [Thompson et al.,
Phys. Rev. Lett. 119, 057203 (2017)]. Turning to low-fields, though the extent
of the experimentally observed broadening is not quite reproduced, we find a
rough correspondence between non-linear spin-wave theory and inelastic neutron
scattering data on both a single-crystal sample, as well as on a powder sample
[Pe\c{c}anha-Antonio et al., Phys. Rev. B, 96, 214415 (2017)]. We conclude with
an outlook on implications for future experimental and theoretical work on
Yb$_2$Ti$_2$O$_7$ and related materials, highlighting the importance of
proximity to the splayed ferromagnet-$\Gamma_5$ phase boundary and its
potential role in intrinsic or extrinsic explanations of the low-field physics
of Yb$_2$Ti$_2$O$_7$.
