Stability and Instability of Coherent Structures
The kink-anti-kink profile of a bacterial colony, the
local deformations of an elastic filament, or the ultra-short
pulses propagating in an optical fiber, are all examples of
coherent structures. Such objects are often described as special
solutions of one or more model partial differential equations, and
the question of their stability, as predicted by these evolution
equations, arises naturally. Modern analytical techniques have
been developed to answer such questions. For instance, the Evans
function is an analytic function which vanishes on the point spectrum
of a linear operator. Asymptotic expansions can sometimes be used to
prove that this function has a zero on the positive real axis,
leading to an instability result. But often, a numerical investigation
of the number of zeros of the Evans function in the right half complex
plane is necessary to obtain complete spectral stability. In the case
of Hamiltonian systems, other techniques are available, which lead to
analytical stability results. Such tools and methods, which
combine analysis and numerical simulations, are being applied to a
variety of evolution partial differential equations, relevant to problems
in physics and biology.
