  Wave equations containing spatial derivatives which are higher than second
order arise naturally in the context of condensed matter systems. The solutions
of such equations contain more than two modes and consequently, the range of
possible interactions between the different modes is significantly enhanced
compared to the two mode case. We develop a framework for analysing the
different mode interactions based on the classical turning points of the
dispersion relation. We then apply this framework to the scattering of deep
water gravity waves with a draining bathtub vortex, a system which constitutes
the analogue of a rotating black hole in the non-dispersive limit. In
particular, we show that the different scattering outcomes are controlled by
the light-ring frequencies, a concept routinely applied in black hole physics,
and two new frequencies which are related to the strength of dispersion. We
find that the frequency range in which the reflected wave is superradiantly
amplified appears as a simple modification to the non-dispersive case. However,
the condition to observe this amplification is complicated by the fact that a
superradiant mode can be reflected back into the system by scattering with one
of the additional modes. We provide estimates for the reflection coefficients
in the full dispersive regime.
