Computing the Evans function using Grassmannians

Abstract

We present a numerical method for computing the pure-point spectrum associated with the linear stability of coherent structures. Our method is based on the Evans function shooting and matching approach. The Grassmann representatives for the stable and unstable manifolds of the spectral problem suffice to construct the Evans function. Our idea is to fix a coordinate patch for the Grassmann representatives of each manifold and numerically compute in that representation. We are thus required to solve a nonlinear Riccati differential equation for each manifold. In practice the method is stable, robust, analytic in the spectral parameter and of complexity bounded by the order of the spectral problem. For large systems it represents a competitive method to that proposed by Humpherys and Zumbrun [21]. We demonstrate this by comparing the two methods in three applications: Boussinesq solitary waves, autocatalytic travelling waves and Ekman boundary layer