PDEBOOK Symmetry properties of solutions of partial differential
equations often translate into special, symmetric solutions
such as self-similar solutions, travelling waves and
radially symmetric solutions, which play a key role in
applications. They are often solutions of ordinary differential
equations with a special structure.
The literature on such equations is ubiquitous but scattered
over the mathematical, engineering, and physics literature.
C.J. van Duijn and L.A. Peletier are writing a textbook
aimed at graduate students in mathematics, physics and engineering,
offering a systematic introduction to these equations and the
properties of their solutions.
GS In 1983 Gray and Scott proposed a system of two reaction-diffusion
equations as a model system for understanding complex dynamics in
chemical reactions, and the understanding of the formation of patterns
such as the birth of multi-bump spikes and travelling fronts.
This system leads to nonlinear eigenvalue problems with several
eigenvalues, involving a variety of (repeated) bifurcations.
In a joint project A. Doelman, L.A. Peletier and T.J. Kaper from
Boston University are studying a sequence of saddle-node bifurcations
leading to a hierarchy of spikes with increasing complexity.
LFBP Free boundary problems (FBP's) for partial differential equations
(PDE's) appear in many applications in the exact sciences. The classical example is the Stefan
problem for water-ice. Other applications involve cell boundaries,
contact lines in thin
film flows, and flame fronts in combustion models.
FBP's may appear as limits of (systems of) reaction-diffusion equations (RDE's)
which have been studied extensively from a dynamical systems viewpoint.
In this project we aim at a theory for FBP's which parallels that of RDE's:
linearisation, Evans functions, rigorous nonlinear stability and bifurcation
analysis, etc. A main application is the thermo-diffusive model for combustion
in dusty,
gaseous mixtures, where the presence of dust accounts for nonlocal terms in
the temperature equation. These are due to radiative effects which have a
strong effect on flame temperatures and speeds.