Evans function and Fredholm determinants

TL;DR

This paper investigates the mathematical relationships between the Evans function, transmission coefficient, and Fredholm determinant in the context of linear stability analysis of traveling wave solutions, providing new theoretical clarifications.

Contribution

It clarifies the equivalence between the Evans function and transmission coefficient, and proves the equivalence of the transmission coefficient and Fredholm determinant for certain operators.

Findings

Established the equivalence between Evans function and transmission coefficient.

Proved the equivalence of transmission coefficient and Fredholm determinant for trace class perturbations.

Provided theoretical insights applicable to stability analysis of traveling waves.

Abstract

We explore the relationship between the Evans function, transmission coefficient and Fredholm determinant for systems of first order linear differential operators on the real line. The applications we have in mind include linear stability problems associated with travelling wave solutions to nonlinear partial differential equations, for example reaction-diffusion or solitary wave equations. The Evans function and transmission coefficient, which are both finite determinants, are natural tools for both analytic and numerical determination of eigenvalues of such linear operators. However, inverting the eigenvalue problem by the free state operator generates a natural linear integral eigenvalue problem whose solvability is determined through the corresponding infinite Fredholm determinant. The relationship between all three determinants has received a lot of recent attention. We focus on the case when the underlying Fredholm operator is a trace class perturbation of the identity. Our new results include: (i) clarification of the sense in which the Evans function and transmission coefficient are equivalent; and (ii) proof of the equivalence of the transmission coefficient and Fredholm determinant, in particular in the case of distinct far fields.