We sketch a proof of global existence and scattering for the defocusing cubic nonlinear Schrödinger equation in for . The proof uses a new estimate of Morawetz type.
We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle . The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov []. Unlike the work of Kuksin [] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our results...
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