Koopman analysis of the periodic Korteweg-de Vries equation
| Autor: | Jeremy P. Parker, Claire Valva |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
Nonlinear Sciences - Exactly Solvable and Integrable Systems
Applied Mathematics Fluid Dynamics (physics.flu-dyn) General Physics and Astronomy FOS: Physical sciences Statistical and Nonlinear Physics Physics - Fluid Dynamics Pattern Formation and Solitons (nlin.PS) Nonlinear Sciences - Pattern Formation and Solitons kdv spectral properties dynamic-mode decomposition systems Exactly Solvable and Integrable Systems (nlin.SI) Mathematical Physics |
| DOI: | 10.48550/arxiv.2211.17119 |
| Popis: | 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. (c) 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|