Stability of solutions to PDEs through the numerical evaluation of the Evans function

Abstract: When considering solutions to PDEs, a fundamental problem is to determine their stability properties. Solutions that are unstable under small perturbations will not be observed experimentally.

Such a stability study gives rise to an eigenvalue problem associated with a differential operator. The Evans function is an object designed to locate the spectrum of differential operators and it has been used numerous times in the past for establishing stability or instability properties of solutions to PDEs. However, it is usually impossible to calculate the Evans function explicitly and different methods to evaluate it numerically have been proposed in the past.

In this talk I will present a numerical method to calculate the Evans function numerically which distinguishes itself by its simplicity. It will be used in two different contexts: in elasticity theory and in combustion theory.