  Ubiquitous nonlinear waves in dispersive media include localized solitons and
extended hydrodynamic states such as dispersive shock waves. Despite their
physical prominence and the development of thorough theoretical and
experimental investigations of each separately, experiments and a unified
theory of solitons and dispersive hydrodynamics are lacking. Here, a general
soliton-mean field theory is introduced and used to describe the propagation of
solitons in macroscopic hydrodynamic flows. Two universal adiabatic invariants
of motion are identified that predict trapping or transmission of solitons by
hydrodynamic states. The result of solitons incident upon smooth expansion
waves or compressive, rapidly oscillating dispersive shock waves is the same,
an effect termed hydrodynamic reciprocity. Experiments on viscous fluid
conduits quantitatively confirm the soliton-mean field theory with broader
implications for nonlinear optics, superfluids, geophysical fluids, and other
dispersive hydrodynamic media.
