  The viscously dominated, low Reynolds' number dynamics of multi-phase,
compacting media can lead to nonlinear, dissipationless/dispersive behavior
when viewed appropriately. In these systems, nonlinear self-steepening competes
with wave dispersion, giving rise to dispersive shock waves (DSWs). Example
systems considered here include magma migration through the mantle as well as
the buoyant ascent of a low density fluid through a viscously deformable
conduit. These flows are modeled by a third-order, degenerate, dispersive,
nonlinear wave equation for the porosity (magma volume fraction) or
cross-sectional area, respectively. Whitham averaging theory for step initial
conditions is used to compute analytical, closed form predictions for the DSW
speeds and the leading edge amplitude in terms of the constitutive parameters
and initial jump height. Novel physical behaviors are identified including
backflow and DSW implosion for initial jumps sufficient to cause gradient
catastrophe in the Whitham modulation equations. Theoretical predictions are
shown to be in excellent agreement with long-time numerical simulations for the
case of small to moderate amplitude DSWs. Verifiable criteria identifying the
breakdown of this modulation theory in the large jump regime, applicable to a
wide class of DSW problems, are presented.
