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.
Jan 14 1998