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.
Highlights
Our goal is to establish the connection between the Evans function, transmission coefficient and Fredholm determinant associated with linear nth order eigenvalue problems on R of the form
We have the following observations on the results above: (i) at the core of the equivalence theorem is, for solutions decaying in the far field, on one hand we have the Fredholm determinant associated with the solvability ofφ = O, while on the other we have the transmission coefficient associated with the solvability ofφ− = φ0−; (ii) we note we have rspa.royalsocietypublishing.org Proc
We have established the equivalence of the Evans function and transmission coefficient for the eigenvalue problem (∂ − A0 − V)Y = O, in the sense that the ratio of the Evans function to the free Evans function is equal to the ratio of the transmission coefficient to the free transmission coefficient
Summary
Our goal is to establish the connection between the Evans function, transmission coefficient and Fredholm determinant associated with linear nth order eigenvalue problems on R of the form (∂ − A0 − V)Y = O. Our goal is to establish a unified picture of the relationship between the Evans function, transmission coefficient and Fredholm determinant. We focus on those systems of first-order operators for which |V|1/2(∂ − A0)−1U|V|1/2 is trace class and the matrix trace of the matrix perturbation potential V is zero. We provide a simple and direct proof that the scaled transmission coefficient equals the Fredholm determinant of |V|1/2(∂ − A0)−1U|V|1/2, assuming it is trace class (see §6).
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