Björn Sandstede
Björn Sandstede
Over the first month or so, I plan to give an introduction to spatial dynamics which will cover the motivation behind this approach and the technical details for carrying it out, starting with simple situations and moving on to more advanced ones.
In many circumstances, one is interested in finding travelling-wave solutions u(x, t) = v(x - ct) say of reaction-diffusion systems ut = uxx + f (u) with x in R. Fortunately, travelling waves v(x - ct) satisfy the ordinary differential equation - cvy = vyy + f (v) which can be analysed using dynamical-systems methods. "Spatial dynamics" refers to the idea of studying more complicated patterns, such as time-periodic solutions, spiral waves, ..., from exactly the same view point, namely by considering the spatial variable x, and not the time t, as the evolution variable. These patterns cannot be captured by ordinary differential equations, but it turns out that many dynamical-systems ideas nevertheless carry over. Spatial dynamics was originally conceived of by Klaus Kirchgässner in 1982 in the context of water waves and further developed by Alexander Mielke in the mid 80's (who applied it in particular to problems in elasticity and the Navier-Stokes equations). This earlier work is concerned with small-amplitude waves. I will concentrate mainly on later work (from the mid 90's onwards) on time-periodic waves of arbitrary amplitude.
