Koopman analysis of the periodic Korteweg-de Vries equation

TL;DR

This paper analytically derives the Koopman eigenfunctions for the periodic Korteweg-de Vries equation using inverse scattering and algebraic geometry, providing a novel exact Koopman analysis for a PDE without a trivial attractor.

Contribution

It presents the first complete Koopman analysis of a non-trivial PDE, specifically the Korteweg-de Vries equation on a periodic domain, using analytical methods.

Findings

Koopman eigenfunctions are derived exactly for the KdV equation.

Results match frequencies obtained by dynamic mode decomposition.

DMD eigenvalues near the imaginary axis are interpreted in this context.

Abstract

The eigenspectrum of the Koopman operator enables the decomposition of nonlinear dynamics into a sum of nonlinear functions of the state space with purely exponential and sinusoidal time dependence. For a limited number of dynamical systems, it is possible to find these Koopman eigenfunctions exactly and analytically. Here, this is done for the Korteweg-de Vries equation on a periodic interval, using the periodic inverse scattering transform and some concepts of algebraic geometry. To the authors' knowledge, this is the first complete Koopman analysis of a partial differential equation which does not have a trivial global attractor. The results are shown to match the frequencies computed by the data-driven method of dynamic mode decomposition (DMD). We demonstrate that in general DMD gives a large number of eigenvalues near the imaginary axis, and show how these should be interpretted in this setting.