  The eigenspectrum of the Koopman operator enables the decomposition of
nonlinear dynamics into a sum of nonlinear functions of the state space with
purely exponential and sinusoidal time dependence. For a limited number of
dynamical systems, it is possible to find these Koopman eigenfunctions exactly
and analytically. Here, this is done for the Korteweg-de Vries equation on a
periodic interval, using the periodic inverse scattering transform and some
concepts of algebraic geometry. To the authors' knowledge, this is the first
complete Koopman analysis of a partial differential equation which does not
have a trivial global attractor. The results are shown to match the frequencies
computed by the data-driven method of dynamic mode decomposition (DMD). We
demonstrate that in general DMD gives a large number of eigenvalues near the
imaginary axis, and show how these should be interpretted in this setting.
