• DOI:10.1007/s40598-020-00162-8
• Corpus ID: 211677843

Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey

@article{Modin2020IntegrabilityOP,
  title={Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey},
  author={Klas Modin and Milo Viviani},
  journal={Arnold Mathematical Journal},
  year={2020},
  volume={7},
  pages={357 - 385}
}

• K. Modin, M. Viviani
• Published 2 March 2020
• Mathematics, Physics
• Arnold Mathematical Journal

Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document}, 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.