Numerical Continuation and Bifurcation in Nonlinear PDEs: Stability, invasion and wavetrains in the Swift-Hohenberg equation

Abstract

We discuss some aspects of numerical continuation and bifurcation for partial differential equations, specifically pattern formation and coherent structures. For the sake of clarity we focus on wavetrains, stability and associated invasion processes in the paradigmatic cubic Swift-Hohenberg equation (SHE). We do not aim at a review of numerical continuation for PDE or pattern formation in SHE in any generality, rather our goal is to provide an entry point for interested students and colleagues to the application of continuation methods. We provide access to numerical implementations and hope that our presentation provides an introductory guideline that can also be used for teaching.