  Rayleigh-Taylor instability (RTI) occurs in a range of industrial and natural
processes. Whereas the vast majority of existing studies have considered
constant acceleration, RTI is in most instances driven by variable
acceleration. Here we focus on RTI driven by acceleration with a power-law
time-dependence, and by applying a group theoretic method find solutions to
this classical nonlinear boundary value problem. We deduce that the dynamics is
dominated by the acceleration term and that the solutions depend critically on
the time dependence for values of the acceleration exponent greater than $-2$.
We find that in the early-time dynamics, the RTI growth-rate depends on the
acceleration parameters and initial conditions. For the later-time dynamics, we
link the interface dynamics with an interfacial shear function, and find a
continuous family of regular asymptotic solutions and invariant properties of
nonlinear RTI. The essentially interfacial and multi-scale character of the
dynamics is also demonstrated. The velocity field is potential in the bulk, and
vortical structures appear at the interface due to interfacial shear. The
multi-scale character becomes clear from the invariance properties of the
dynamics. We also achieve excellent agreement with existing observations and
elaborate new benchmarks for future experimental work.
