---
OA_place: repository
OA_type: green
_id: '22064'
abstract:
- lang: eng
  text: Using the two-dimensional nonlinear Schrödinger equation as a model example,
    we present a general method for recovering the nonlinearity of a nonlinear dispersive
    equation from its small-data scattering behavior. We prove that under very mild
    assumptions on the nonlinearity, the wave operator uniquely determines the nonlinearity,
    as does the scattering map. Evaluating the scattering map on well-chosen initial
    data, we reduce the problem to an inverse convolution problem, which we solve
    by means of an application of the Beurling–Lax Theorem.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Jason
  full_name: Murphy, Jason
  last_name: Murphy
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  ama: Killip R, Murphy J, Vişan M. The scattering map determines the nonlinearity.
    <i>Proceedings of the American Mathematical Society</i>. 2023;151(6):2543-2557.
    doi:<a href="https://doi.org/10.1090/proc/16297">10.1090/proc/16297</a>
  apa: Killip, R., Murphy, J., &#38; Vişan, M. (2023). The scattering map determines
    the nonlinearity. <i>Proceedings of the American Mathematical Society</i>. American
    Mathematical Society. <a href="https://doi.org/10.1090/proc/16297">https://doi.org/10.1090/proc/16297</a>
  chicago: Killip, Rowan, Jason Murphy, and Monica Vişan. “The Scattering Map Determines
    the Nonlinearity.” <i>Proceedings of the American Mathematical Society</i>. American
    Mathematical Society, 2023. <a href="https://doi.org/10.1090/proc/16297">https://doi.org/10.1090/proc/16297</a>.
  ieee: R. Killip, J. Murphy, and M. Vişan, “The scattering map determines the nonlinearity,”
    <i>Proceedings of the American Mathematical Society</i>, vol. 151, no. 6. American
    Mathematical Society, pp. 2543–2557, 2023.
  ista: Killip R, Murphy J, Vişan M. 2023. The scattering map determines the nonlinearity.
    Proceedings of the American Mathematical Society. 151(6), 2543–2557.
  mla: Killip, Rowan, et al. “The Scattering Map Determines the Nonlinearity.” <i>Proceedings
    of the American Mathematical Society</i>, vol. 151, no. 6, American Mathematical
    Society, 2023, pp. 2543–57, doi:<a href="https://doi.org/10.1090/proc/16297">10.1090/proc/16297</a>.
  short: R. Killip, J. Murphy, M. Vişan, Proceedings of the American Mathematical
    Society 151 (2023) 2543–2557.
das_tickbox: '1'
date_created: 2026-06-19T08:12:42Z
date_published: 2023-06-01T00:00:00Z
date_updated: 2026-06-30T07:08:20Z
day: '01'
doi: 10.1090/proc/16297
extern: '1'
external_id:
  arxiv:
  - '2207.02414'
intvolume: '       151'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2207.02414
mathsc:
- 35Q55
month: '06'
oa: 1
oa_version: Preprint
page: 2543-2557
publication: Proceedings of the American Mathematical Society
publication_identifier:
  eissn:
  - 1088-6826
  issn:
  - 0002-9939
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: The scattering map determines the nonlinearity
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 151
year: '2023'
...