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Small divisors and free vibrations in the n-vortex filament problem

Banff International Research Station for Mathematical Innovation and Discovery

In this talk we discuss families of homographic standing waves appearing from n vortex filaments rotating uniformly at a central configuration. The solution of the filaments are time periodic with periodic boundary conditions, i.e. this is a small divisor problem for a Hamiltonian partial differential equation which requires techniques related to KAM theory. In this case the Nash-Moser method gives rise to a family of solutions over a Cantor set of parameters. On the other hand, we show that when the relation between temporal and spatial periods is fixed at certain rational numbers, the contraction mapping theorem gives existence of an infinite number of families of standing waves that bifurcate from these configurations.

Mathematics; Dynamical systems and ergodic theory; Partial differential equations

video/mp4

Author affiliation: UNAM

2019-12-10

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72636

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/