 0830 Fritz Gesztesy, Yuri Latushkin, and Kevin Zumbrun
 Derivatives of (Modified) Fredholm Determinants and Stability of
Standing and Traveling Waves
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Feb 16, 08

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Abstract. Continuing a line of investigation initiated in \cite{GLM07}
exploring the connections between Jost and Evans functions and
(modified) Fredholm determinants of BirmanSchwinger type integral
operators, we here examine the stability index, or sign of the
first nonvanishing derivative at frequency zero of the
characteristic determinant, an object that has found considerable
use in the study by Evans function techniques of stability of
standing and traveling wave solutions of partial differential
equations (PDE) in one dimension. This leads us to the derivation
of general perturbation expansions for analyticallyvarying
modified Fredholm determinants of abstract operators. Our main
conclusion, similarly in
the analysis of the determinant itself, is that the derivative of
the characteristic Fredholm determinant may be efficiently
computed from first principles for integral operators with
semiseparable integral kernels, which
include in particular the general onedimensional case, and for
sums thereof, which latter possibility appears to offer
applications in the multidimensional case.
A second main result is to show that the multidimensional
characteristic Fredholm determinant is the renormalized limit
of a sequence of Evans functions
defined in \cite{LPSS00} on successive Galerkin subspaces,
giving a natural extension of the onedimensional results
of \cite{GLM07} and
answering a question of \cite{N07} whether this sequence
might possibly converge (in general, no, but with renormalization, yes).
Convergence is useful in practice for numerical error control
and acceleration.
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