  The propagation of localized solitons in the presence of large-scale waves is
a fundamental problem, both physically and mathematically, with applications in
fluid dynamics, nonlinear optics and condensed matter physics. Here, the
evolution of a soliton as it interacts with a rarefaction wave or a dispersive
shock wave, examples of slowly varying and rapidly oscillating dispersive mean
fields, for the Korteweg-de Vries equation is studied. Step boundary conditions
give rise to either a rarefaction wave (step up) or a dispersive shock wave
(step down). When a soliton interacts with one of these mean fields, it can
either transmit through (tunnel) or become embedded (trapped) inside, depending
on its initial amplitude and position. A comprehensive review of three separate
analytical approaches is undertaken to describe these interactions. First, a
basic soliton perturbation theory is introduced that is found to capture the
solution dynamics for soliton-rarefaction wave interaction in the small
dispersion limit. Next, multiphase Whitham modulation theory and its finite-gap
description are used to describe soliton-rarefaction wave and
soliton-dispersive shock wave interactions. Lastly, a spectral description and
an exact solution of the initial value problem is obtained through the Inverse
Scattering Transform. For transmitted solitons, far-field asymptotics reveal
the soliton phase shift through either type of wave mentioned above. In the
trapped case, there is no proper eigenvalue in the spectral description,
implying that the evolution does not involve a proper soliton solution. These
approaches are consistent, agree with direct numerical simulation, and
accurately describe different aspects of solitary wave-mean field interaction
