  We consider two non-interacting two-level emitters that are coupled weakly to
a one-dimensional non-linear wave guide. Due to the Kerr-like non-linearity,
the wave guide considered supports -- in addition to the scattering continuum
-- a two-body bound state. As such, the wave guide models a bath with
non-trivial mode structure. Solving the time-dependent Schr\"odinger equation,
the radiation dynamics of the two emitters, initially prepared in their excited
states, is presented. Changing the emitter frequency such that the two-emitter
energy is in resonance with one of the two-body bound states, radiation
dynamics ranging from exponential decay to fractional populations to Rabi
oscillations is observed. Along with the detuning, the dependence on the
separation of the two emitters is investigated. Approximate reduced Hilbert
space formulations, which result in effective emitter-separation and momentum
dependent interactions, elucidate the underlying physical mechanisms and
provide an avenue to showcase the features that would be absent if the
one-dimensional wave guide did not contain a non-linearity. Our theoretical
findings apply to a number of experimental platforms and the predictions can be
tested with state-of-the-art technology. In addition, the weak-coupling
Schr\"odinger equation based results provide critical guidance for the
development of master equation approaches.
