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Updated: Aug 1, 2025

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations
Published on: October 12, 2019
Koopman analysis of the periodic Korteweg-de Vries equation
Jeremy P Parker1, Claire Valva2
1Emergent Complexity in Physical Systems Laboratory (ECPS), École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.
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 interpreted in this setting.
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