Koopman analysis of Burgers equation

TL;DR

This paper derives explicit Koopman modes and eigenfunctions for Burgers equation, demonstrating the complexities of Koopman analysis for nonlinear PDEs and evaluating DMD's effectiveness in capturing flow structures.

Contribution

First explicit Koopman decomposition for a nonlinear PDE, revealing dependencies of modes on initial conditions and eigenfunction degeneracies.

Findings

Koopman modes are linearly dependent and cannot be fitted without eigenfunctions.

Eigenvalues are highly degenerate, affecting mode stability.

DMD has limitations in extracting nonlinear coherent structures.

Abstract

The emergence of Dynamic Mode Decomposition (DMD) as a practical way to attempt a Koopman mode decomposition of a nonlinear PDE presents exciting prospects for identifying invariant sets and slowly decaying transient structures buried in the PDE dynamics. However, there are many subtleties in connecting DMD to Koopman analysis and it remains unclear how realistic Koopman analysis is for complex systems such as the Navier-Stokes equations. With this as motivation, we present here a full Koopman decomposition for the velocity field in Burgers equation by deriving explicit expressions for the Koopman modes and eigenfunctions - the first time this has been done for a nonlinear PDE. The decomposition highlights the fact that different observables can require different subsets of Koopman eigenfunctions to express them and presents a nice example where: (i) the Koopman modes are linearly dependent and so cannot be fit a posteriori to snapshots of the flow without knowledge of the Koopman eigenfunctions; and (ii) the Koopman eigenvalues are highly degenerate which means that computed Koopman modes become initial-condition dependent. As way of illustration, we discuss the form of the Koopman expansion with various initial conditions and assess the capability of DMD to extract the decaying nonlinear coherent structures in run-down simulations.