  We obtain the effective theory for the non-linear dynamics of black
branes---both neutral and charged, in asymptotically flat or Anti-deSitter
spacetimes---to leading order in the inverse-dimensional expansion. We find
that black branes evolve as viscous fluids, but when they settle down they are
more naturally viewed as solutions of an elastic soap-bubble theory. The two
views are complementary: the same variable is regarded in one case as the
energy density of the fluid, in the other as the deformation of the elastic
membrane. The large-D theory captures finite-wavelength phenomena beyond the
conventional reach of hydrodynamics. For asymptotically flat charged black
branes (either Reissner-Nordstrom or p-brane-charged black branes) it yields
the non-linear evolution of the Gregory-Laflamme instability at large D and its
endpoint at stable non-uniform black branes. For Reissner-Nordstrom AdS black
branes we find that sound perturbations do not propagate (have purely imaginary
frequency) when their wavelength is below a certain charge-dependent value. We
also study the polarization of black branes induced by an external electric
field.
