Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2

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.