### Björn Sandstede

## Stability of travelling waves

In this article, an overview on various aspects of the stability of
travelling-wave solutions to partial differential equations is given.

The first part will deal with tracking isolated eigenvalues using the
Evans function. The Evans function is introduced and its main
properties are reviewed. Furthermore, recent results on the analytic
extension of the Evans function across the essential spectrum are
given. This extension is particularly important when the PDE is either
conservative or a dissipative perturbation of a conservative system.
While the Evans function appears to be primarily a tool for ordinary
differential equations, it can actually be used for many non-local
PDE's. Moreover, it may be possible to construct an Evans function for
elliptic problem on infinite cylinders.

The second part is then concerned with the stability of multiple
pulses (i.e. multi-bump pulses or **n**-pulses). These are travelling
waves consisting of several widely spaced, concatenated copies of a
primary pulse. The spectrum of the linearization about an **n**-pulse is
essentially given by **n** copies of the spectrum of the primary pulse.
Since zero is always an eigenvalue due to translation invariance,
there are in particular (at least) **n** eigenvalues near zero. The
issue is then to calculate these eigenvalues and to determine whether
they all move into the left half-plane. We focus on analytical and
topological methods to tackle this problem. In addition, results on
the interaction of widely spaced pulses are reviewed.

In the third part, stability and instability results for conservative
PDE's are considered briefly. In particular, we review the technique
of constraint minimization.

These parts will be accompanied with examples from nonlinear optics,
mathematical biology (FitzHugh-Nagumo equation) and possibly other
areas.

*
Björn Sandstede*

Jan 14 1998