  There is a well explored relationship between quantum mechanical scattering
from a potential and the Korteweg-de Vries (KdV) equation of fluid dynamics: if
the potential is 'evolved' according to the KdV equation then it will have the
same reflectivity and transmissivity as a function of energy, for each snapshot
in time. In this work we explore this connection in optics, where the
permittivity plays the role of the potential. We begin by deriving the
relationship between the Helmholtz equation and the KdV equation in terms of
the current induced in a material when a permittivity profile is changed
slightly. It is then shown that the KdV equation can be used to design a
plethora of bounded complex potentials that are relfectionless from both sides
for all angles of incidence, and planar periodic media that exhibit a real
Bloch vector for all angles of propagation. Finally we apply the KdV equation
to reduce the reflection of a wave from an interface between two media of
differing refractive indices.
