On Koopman Operator for Burgers' Equation

TL;DR

This paper explicitly derives the Koopman decomposition for Burgers' equation, identifying eigenvalues and eigenfunctionals, proving convergence, and constructing all modes, including eigenspaces, with numerical comparisons to DMD.

Contribution

It provides a complete explicit Koopman decomposition for Burgers' equation, including all modes and eigenspaces, extending previous partial results.

Findings

Explicit Koopman decomposition for Burgers' equation derived.

Convergence of the decomposition established for small and regular data.

Numerical comparison between Koopman eigenvalues and DMD eigenvalues conducted.

Abstract

We consider the flow of Burgers' equation on an open set of (small) functions in $L^2([0,1])$. We derive explicitly the Koopman decomposition of the Burgers' flow. We identify the frequencies and the coefficients of this decomposition as eigenvalues and eigenfunctionals of the Koopman operator. We prove the convergence of the Koopman decomposition for $t>0$ for small Cauchy data, and up to $t=0$ for regular Cauchy data. The convergence up to $t=0$} leads to a `completeness' property for the basis of Koopman modes. We construct all modes and eigenfunctionals, including the eigenspaces involved in geometric multiplicity. This goes beyond the summation formulas provided by (Page & Kerswell, 2018), where only one term per eigenvalue was given. A numeric illustration of the Koopman decomposition is given and the Koopman eigenvalues compared to the eigenvalues of a Dynamic Mode Decomposition (DMD).