It is well known that a material interface between fluids of different density is unstable under an impulsive acceleration of a shock wave. Fingers develop at the interfaces. The basic structures of the fingers are spikes
and bubbles. A spike (bubble) is portion of the heavy (light) fluid penetrating into light (heavy) fluid.  The existing linear theory and the impulsive model provide qualitatively incorrect predictions for the growth of the fingers at late time.  We present a nonlinear theory for the growth of the spikes and bubbles of this unstable system. Our theory is based on the methods of Pade approximation and asymptotic matching and contains no adjustable parameters. Our theoretical predictions are in remarkably good agreement with the results from full numerical simulations in both two and three dimensions and with the experimental data over the entire time range of the simulations and experiments.  Our theory shows that the unstable system goes through a transition from a compressible, linear one at early time, to a nonlinear, incompressible one at late time.