Journal article
Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2
American journal of mathematics, Vol.143(2), pp.613-680
2021
DOI: 10.1353/ajm.2021.0014
Abstract
We prove that solutions of the cubic nonlinear Schrödinger equation on R2 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. Complement-ing 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 R2. 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.
Details
- Title: Subtitle
- Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2
- Creators
- Rowan KillipMonica VisanXiaoyi Zhang
- Resource Type
- Journal article
- Publication Details
- American journal of mathematics, Vol.143(2), pp.613-680
- Publisher
- Johns Hopkins University Press
- DOI
- 10.1353/ajm.2021.0014
- ISSN
- 0002-9327
- eISSN
- 1080-6377
- Language
- English
- Date published
- 2021
- Academic Unit
- Mathematics
- Record Identifier
- 9984240776102771
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