  The Einstein equations describing the black-brane dynamics both in Minkowski
and AdS background were recently recast in the form of coupled diffusion
equations in the large-$D$(imension) limit. Using such results in the
literature, we formulate a higher-order perturbation theory of black branes in
time domain and present the general form of solutions for arbitrary initial
conditions. For illustrative purposes, the solutions up to the first or second
order are explicitly written down for several kind of initial conditions, such
as a Gaussian wave packet, shock wave, and rather general superposed sinusoidal
waves. These could be the first examples describing the non-trivial evolution
of black-brane horizons in time domain. In particular, we learn some
interesting aspects of black-brane dynamics such as the Gregory-Laflamme (GL)
instability and non-equilibrium steady state (NESS). The formalism presented
here would be applicable to the analysis of various black branes and their
holographically dual field theories.
