Abstract
We explore the relationship between the Evans function, transmission coefficient and Fredholm determinant for systems of firstorder 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 reactiondiffusion 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 freestate 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.
Original language  English 

Article number  20140597 
Journal  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 
Volume  471 
Issue number  2174 
DOIs  
Publication status  Published  2015 
Keywords
 Evans function
 Fredholm determinant
 Travelling waves
ASJC Scopus subject areas
 General Mathematics
 General Engineering
 General Physics and Astronomy
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Simon John A. Malham
 School of Mathematical & Computer Sciences  Associate Professor
 School of Mathematical & Computer Sciences, Mathematics  Associate Professor
Person: Academic (Research & Teaching)