  Recent numerical work on the Zabusky--Kruskal experiment has revealed,
amongst other things, the existence of hidden solitons in the wave profile.
Here, using Osborne's nonlinear Fourier analysis, which is based on the
periodic, inverse scattering transform, the hidden soliton hypothesis is
corroborated, and the \emph{exact} number of solitons, their amplitudes and
their reference level is computed. Other "less nonlinear" oscillation modes,
which are not solitons, are also found to have nontrivial energy contributions
over certain ranges of the dispersion parameter. In addition, the reference
level is found to be a non-monotone function of the dispersion parameter.
Finally, in the case of large dispersion, we show that the one-term nonlinear
Fourier series yields a very accurate approximate solution in terms of Jacobian
elliptic functions.
