Applied Mathematics seminar | Issa Karambal, Evans function and Fredholm determinants

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.