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Symplectic non-squeezing for the discrete nonlinear Schrodinger equation

Banff International Research Station for Mathematical Innovation and Discovery

The celebrated Gromov's non-squeezing theorem of 1985 says that the unit ball $B^n$ in $C^n$ can be symplectically embedded in the "cylinder" $rB^1 \times C^{n-1}$ of radius r only if $r\ge 1$. Hamiltonian differential equations provide examples of symplectic transformations in infinite dimension. Known results on the non-squeezing property in Hilbert spaces cover compact perturbations of linear symplectic transformations and several specific non-linear PDEs, including the periodic Korteweg - de Vries equation and the periodic cubic Schr\"odinger equation. We prove a new version of the non-squeezing theorem for Hilbert spaces. We apply the result to the discrete nonlinear Schr\"odinger equation. This work is joint with Alexander Sukhov.

Mathematics; Several complex variables and analytic spaces; Potential theory; Global analysis

video/mp4

Author affiliation: University of Illinois at Urbana-Champaign

2017-02-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59620

Unreviewed

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