  The large-time behavior of solutions to Burgers equation with small viscosity
is described using invariant manifolds. In particular, a geometric explanation
is provided for a phenomenon known as metastability, which in the present
context means that solutions spend a very long time near the family of
solutions known as diffusive N-waves before finally converging to a stable
self-similar diffusion wave. More precisely, it is shown that in terms of
similarity, or scaling, variables in an algebraically weighted $L^2$ space, the
self-similar diffusion waves correspond to a one-dimensional global center
manifold of stationary solutions. Through each of these fixed points there
exists a one-dimensional, global, attractive, invariant manifold corresponding
to the diffusive N-waves. Thus, metastability corresponds to a fast transient
in which solutions approach this "metastable" manifold of diffusive N-waves,
followed by a slow decay along this manifold, and, finally, convergence to the
self-similar diffusion wave.
