  Dispersive shock waves and solitons are fundamental nonlinear excitations in
dispersive media, but dispersive shock wave studies to date have been severely
constrained. Here we report on a novel dispersive hydrodynamics testbed: the
effectively frictionless dynamics of interfacial waves between two high
contrast, miscible, low Reynolds' number Stokes fluids. This scenario is
realized by injecting from below a lighter, viscous fluid into a column filled
with high viscosity fluid. The injected fluid forms a deformable pipe whose
diameter is proportional to the injection rate, enabling precise control over
the generation of symmetric interfacial waves. Buoyancy drives nonlinear
interfacial self-steepening while normal stresses give rise to dispersion of
interfacial waves. Extremely slow mass diffusion and mass conservation imply
that the interfacial waves are effectively dissipationless. This enables high
fidelity observations of large amplitude dispersive shock waves in this
spatially extended system, found to agree quantitatively with a nonlinear wave
averaging theory. Furthermore, several highly coherent phenomena are
investigated including dispersive shock wave backflow, the refraction or
absorption of solitons by dispersive shock waves, and the multi-phase merging
of two dispersive shock waves. The complex, coherent, nonlinear mixing of
dispersive shock waves and solitons observed here are universal features of
dissipationless, dispersive hydrodynamic flows.
