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Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 143, Number 2, April 2021
- pp. 613-680
- 10.1353/ajm.2021.0014
- Article
- Additional Information
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.