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

TL;DR

This paper investigates the derivatives of Fredholm determinants related to stability analysis of waves in PDEs, deriving perturbation expansions and showing convergence of multi-dimensional determinants via Evans functions.

Contribution

It introduces general perturbation expansions for modified Fredholm determinants and demonstrates their efficient computation for integral operators with semi-separable kernels, extending stability analysis to multi-dimensional cases.

Findings

Derivative of Fredholm determinant can be computed from first principles.

Multi-dimensional Fredholm determinant is the renormalized limit of Evans functions.

Convergence of Evans functions sequence aids in numerical stability analysis.

Abstract

Continuing a line of investigation initiated in [11] 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 [23] on successive Galerkin subspaces, giving a natural extension of the one-dimensional results of [11] and answering a question of [27] whether this sequence might possibly converge (in general, no, but with renormalization, yes). Convergence is useful in practice for numerical error control and acceleration.