Derivatives of (Modified) Fredholm Determinants and Stability of Standing and Traveling Waves

Details Derivatives of (Modified) Fredholm Determinants and Stability of Standing and Traveling Waves Derivatives of (Modified) Fredholm Determinants and Stability of Standing and Traveling Waves

2008-02-12

arxiv.org/abs/0802.1665v1

Mathematics

Analysis of PDEs

47 pp

Scientific paper

Continuing a line of investigation initiated in \cite{GLM07} exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman--Schwinger 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 analytically-varying 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 semi-separable integral kernels, which include in particular the general one-dimensional case, and for sums thereof, which latter possibility appears to offer applications in the multi-dimensional case. A second main result is to show that the multi-dimensional 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 one-dimensional 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.

Affiliated with

Also associated with

No associations

Rating

Derivatives of (Modified) Fredholm Determinants and Stability of Standing and Traveling Waves does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Derivatives of (Modified) Fredholm Determinants and Stability of Standing and Traveling Waves, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Derivatives of (Modified) Fredholm Determinants and Stability of Standing and Traveling Waves will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-108638