  We study in detail the scattering of light on a soliton propagating in a
waveguide which has been proposed as an experimental system in which one could
observe the analogue Hawking effect. When not applying the rotating wave
approximation, we show that the linearized wave equation governing
perturbations has the same structure as that governing phonon propagation in an
atomic Bose condensate. By taking into account the full dispersion relation, we
then show that the scattering coefficients encoding the production of photon
pairs are amplified by a resonance effect related to the modulation instability
occurring in the presence of a continuous wave. When using a realistic example
of a silicon nitride waveguide on a silica substrate, we find that a soliton of
duration 10 fs would spontaneously emit about one photon pair for every cm it
travels, which makes the effect readily observable. This result is confirmed by
numerically solving the equation encoding the Kerr nonlinearity and governing
the evolution of the full field (soliton plus perturbations). We discuss the
link with previous works devoted to the analogue Hawking effect where the pair
creation rates were about six orders of magnitude smaller.
