  In all of the diverse areas of science where waves play an important role,
one of the most fundamental solutions of the corresponding wave equation is a
stationary wave with constant intensity. The most familiar example is that of a
plane wave propagating in free space. In the presence of any Hermitian
potential, a wave's constant intensity is, however, immediately destroyed due
to scattering. Here we show that this fundamental restriction is conveniently
lifted when working with non-Hermitian potentials. In particular, we present a
whole new class of waves that have constant intensity in the presence of linear
as well as of nonlinear inhomogeneous media with gain and loss. These solutions
allow us to study, for the first time, the fundamental phenomenon of modulation
instability in an inhomogeneous environment. Our results pose a new challenge
for the experiments on non-Hermitian scattering that have recently been put
forward.
