@article{22035,
  abstract     = {We prove that solutions of the cubic nonlinear Schr\"odinger equation on $\Bbb{R}^2$ can be approximated by a finite-dimensional Hamiltonian system, uniformly on bounded sets of initial data. This is despite the wealth of non-compact symmetries: scaling, translation, and Galilei boosts.

Complementing this approximation result, we show that all solutions of the finite-dimensional Hamiltonian system we use can be approximated by the full PDE.

A key ingredient in these results is the development of a general methodology for transfering uniform global space-time bounds to suitable Fourier truncations of dispersive PDE models.

As an application, we prove symplectic non-squeezing (in the sense of Gromov) for the cubic NLS on $\Bbb{R}^2$. This is the first symplectic non-squeezing result for a Hamiltonian PDE in infinite volume. It is also the first unconditional symplectic non-squeezing result in a scaling-critical setting.

Finally, we discuss implications of non-squeezing on the nature of scattering.},
  author       = {Killip, Rowan and Visan, Monica and Zhang, Xiaoyi},
  issn         = {1080-6377},
  journal      = {American Journal of Mathematics},
  number       = {2},
  pages        = {613--680},
  publisher    = {Johns Hopkins University Press},
  title        = {{Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2}},
  doi          = {10.1353/ajm.2021.0014},
  volume       = {143},
  year         = {2021},
}