Determining the stability of solutions is an important aspect
of the analysis of an applied model, because it is typically the stable
solutions that are observed in practice. When studying a PDE model,
several mathematical issues can arise due to the infinite dimensional
nature of the problem. These include the presence of a continuous
component of the spectrum and a lack of a spectral gap. An explanation of
these issues will be given and techniques for overcoming them will be
presented. In particular, the method of scaling variables will be
introduced, which allows one to open up a gap in the spectrum and
construct invariant manifolds. These manifolds can then be used to
determine the nonlinear stability of the solution of interest.