  Thin viscous liquid films driven by capillarity are well described in the
lubrication theory through the thin film equation. In this article, we present
an analytical solution of this equation for a particular initial profile: a
stepped perturbation. This initial condition allows a linearization of the
problem making it amenable to Fourier analysis. The solution is obtained and
characterized. As for a temperature step in the heat equation, self-similarity
of the first kind of the full evolution is demonstrated and a long-term
expression for the excess free energy is derived. In addition, hydrodynamical
fields are described. The solution is then compared to experimental profiles
from a model system: a polystyrene nanostep above the glass transition
temperature which flows due to capillarity. The excellent agreement enables a
precise measurement of the capillary velocity for this polymeric liquid,
without involving any numerical simulation. More generally, as these results
hold for any viscous system driven by capillarity, the present solution may
provide a useful tool in hydrodynamics of thin viscous films.
