---
OA_place: repository
OA_type: green
_id: '22035'
abstract:
- lang: eng
  text: "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.\r\n\r\nComplementing this approximation
    result, we show that all solutions of the finite-dimensional Hamiltonian system
    we use can be approximated by the full PDE.\r\n\r\nA 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.\r\n\r\nAs 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.\r\n\r\nFinally, we discuss
    implications of non-squeezing on the nature of scattering."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: Xiaoyi
  full_name: Zhang, Xiaoyi
  last_name: Zhang
citation:
  ama: Killip R, Vişan M, Zhang X. Finite-dimensional approximation and non-squeezing
    for the cubic nonlinear Schrödinger equation on ℝ2. <i>American Journal of Mathematics</i>.
    2021;143(2):613-680. doi:<a href="https://doi.org/10.1353/ajm.2021.0014">10.1353/ajm.2021.0014</a>
  apa: Killip, R., Vişan, M., &#38; Zhang, X. (2021). Finite-dimensional approximation
    and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. <i>American
    Journal of Mathematics</i>. Johns Hopkins University Press. <a href="https://doi.org/10.1353/ajm.2021.0014">https://doi.org/10.1353/ajm.2021.0014</a>
  chicago: Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Finite-Dimensional Approximation
    and Non-Squeezing for the Cubic Nonlinear Schrödinger Equation on ℝ2.” <i>American
    Journal of Mathematics</i>. Johns Hopkins University Press, 2021. <a href="https://doi.org/10.1353/ajm.2021.0014">https://doi.org/10.1353/ajm.2021.0014</a>.
  ieee: R. Killip, M. Vişan, and X. Zhang, “Finite-dimensional approximation and non-squeezing
    for the cubic nonlinear Schrödinger equation on ℝ2,” <i>American Journal of Mathematics</i>,
    vol. 143, no. 2. Johns Hopkins University Press, pp. 613–680, 2021.
  ista: Killip R, Vişan M, Zhang X. 2021. Finite-dimensional approximation and non-squeezing
    for the cubic nonlinear Schrödinger equation on ℝ2. American Journal of Mathematics.
    143(2), 613–680.
  mla: Killip, Rowan, et al. “Finite-Dimensional Approximation and Non-Squeezing for
    the Cubic Nonlinear Schrödinger Equation on ℝ2.” <i>American Journal of Mathematics</i>,
    vol. 143, no. 2, Johns Hopkins University Press, 2021, pp. 613–80, doi:<a href="https://doi.org/10.1353/ajm.2021.0014">10.1353/ajm.2021.0014</a>.
  short: R. Killip, M. Vişan, X. Zhang, American Journal of Mathematics 143 (2021)
    613–680.
das_tickbox: '1'
date_created: 2026-06-19T07:43:41Z
date_published: 2021-04-01T00:00:00Z
date_updated: 2026-06-22T12:56:27Z
day: '01'
doi: 10.1353/ajm.2021.0014
extern: '1'
external_id:
  arxiv:
  - '1606.07738'
intvolume: '       143'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1606.07738
month: '04'
oa: 1
oa_version: Preprint
page: 613-680
publication: American Journal of Mathematics
publication_identifier:
  eissn:
  - 1080-6377
publication_status: published
publisher: Johns Hopkins University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Finite-dimensional approximation and non-squeezing for the cubic nonlinear
  Schrödinger equation on ℝ2
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 143
year: '2021'
...