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Integrability of point-vortex dynamics via symplectic reduction: a survey
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abstract
Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on 2-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 $N=2$, $3$, or $4$ point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any 2-dimensional manifold; 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 2-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.
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Cited by 1 Pith paper
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On the collapse of three point vortices on surfaces
Three-vortex collapse is self-similar on the plane and sphere, absent on the hyperbolic plane for any analytic distance, but nearly self-similar collapse exists on arbitrary smooth surfaces.
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