  A "metastable solution" to a differential equation typically refers to a
family of solutions for which nearby initial data converges to the family much
faster than evolution along the family. Metastable families have been observed
both experimentally and numerically in various contexts, they are believed to
be particularly relevant for organizing the dynamics of fluid flows. In this
work we propose a candidate metastable family for the Burgers equation with
periodic boundary conditions. Our choice of family is motivated by our
numerical experiments. We furthermore explain the metastable behavior of the
family without reference to the Cole--Hopf transformation, but rather by
linearizing the Burgers equation about the family and analyzing the spectrum of
the resulting operator. We hope this may make the analysis more readily
transferable to more realistic systems like the Navier--Stokes equations. Our
analysis is motivated by ideas from singular perturbation theory and Melnikov
theory.
