  The periodic, inverse scattering transform (PIST) is a powerful analytical
tool in the theory of integrable, nonlinear evolution equations. Osborne
pioneered the use of the PIST in the analysis of data form inherently nonlinear
physical processes. In particular, Osborne's so-called nonlinear Fourier
analysis has been successfully used in the study of waves whose dynamics are
(to a good approximation) governed by the Korteweg--de Vries equation. In this
paper, the mathematical details and a new application of the PIST are
discussed. The numerical aspects of and difficulties in obtaining the nonlinear
Fourier (i.e., PIST) spectrum of a physical data set are also addressed. In
particular, an improved bracketing of the "spectral eigenvalues" (i.e., the
+/-1 crossings of the Floquet discriminant) and a new root-finding algorithm
for computing the latter are proposed. Finally, it is shown how the PIST can be
used to gain insightful information about the phenomenon of soliton-induced
acoustic resonances, by computing the nonlinear Fourier spectrum of a data set
from a simulation of internal solitary wave generation and propagation in the
Yellow Sea.
