---
OA_place: repository
OA_type: green
_id: '22021'
abstract:
- lang: eng
  text: "We establish global well-posedness for both the defocusing and\r\nfocusing
    complex-valued modified Korteweg–de Vries equations on the real line\r\nin modulation
    spaces Ms,2p (R), for all 1 \x14 p < 1 and 0 \x14 s < 3/2 − 1/p. We\r\nwill also
    show that such solutions admit global-in-time bounds in these spaces\r\nand that
    equicontinuous sets of initial data lead to equicontinuous ensembles\r\nof orbits.
    Indeed, such information forms a crucial part of our well-posedness\r\nargument."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Saikatul
  full_name: Haque, Saikatul
  last_name: Haque
- 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: Yunfeng
  full_name: Zhang, Yunfeng
  last_name: Zhang
citation:
  ama: Haque S, Killip R, Vişan M, Zhang Y.  Global well-posedness and equicontinuity
    for modified Korteweg–de Vries equations in modulation spaces. <i>Pure and Applied
    Analysis</i>. 2025;7(3):615-637. doi:<a href="https://doi.org/10.2140/paa.2025.7.615">10.2140/paa.2025.7.615</a>
  apa: Haque, S., Killip, R., Vişan, M., &#38; Zhang, Y. (2025).  Global well-posedness
    and equicontinuity for modified Korteweg–de Vries equations in modulation spaces.
    <i>Pure and Applied Analysis</i>. Mathematical Sciences Publishers. <a href="https://doi.org/10.2140/paa.2025.7.615">https://doi.org/10.2140/paa.2025.7.615</a>
  chicago: Haque, Saikatul, Rowan Killip, Monica Vişan, and Yunfeng Zhang. “ Global
    Well-Posedness and Equicontinuity for Modified Korteweg–de Vries Equations in
    Modulation Spaces.” <i>Pure and Applied Analysis</i>. Mathematical Sciences Publishers,
    2025. <a href="https://doi.org/10.2140/paa.2025.7.615">https://doi.org/10.2140/paa.2025.7.615</a>.
  ieee: S. Haque, R. Killip, M. Vişan, and Y. Zhang, “ Global well-posedness and equicontinuity
    for modified Korteweg–de Vries equations in modulation spaces,” <i>Pure and Applied
    Analysis</i>, vol. 7, no. 3. Mathematical Sciences Publishers, pp. 615–637, 2025.
  ista: Haque S, Killip R, Vişan M, Zhang Y. 2025.  Global well-posedness and equicontinuity
    for modified Korteweg–de Vries equations in modulation spaces. Pure and Applied
    Analysis. 7(3), 615–637.
  mla: Haque, Saikatul, et al. “ Global Well-Posedness and Equicontinuity for Modified
    Korteweg–de Vries Equations in Modulation Spaces.” <i>Pure and Applied Analysis</i>,
    vol. 7, no. 3, Mathematical Sciences Publishers, 2025, pp. 615–37, doi:<a href="https://doi.org/10.2140/paa.2025.7.615">10.2140/paa.2025.7.615</a>.
  short: S. Haque, R. Killip, M. Vişan, Y. Zhang, Pure and Applied Analysis 7 (2025)
    615–637.
das_tickbox: '1'
date_created: 2026-06-19T07:30:23Z
date_published: 2025-06-18T00:00:00Z
date_updated: 2026-06-24T13:22:40Z
day: '18'
doi: 10.2140/paa.2025.7.615
extern: '1'
external_id:
  arxiv:
  - '2411.05300'
intvolume: '         7'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2411.05300
mathsc:
- 35Q53
- 35Q55
- 37K10
month: '06'
oa: 1
oa_version: Preprint
page: 615-637
publication: Pure and Applied Analysis
publication_identifier:
  eissn:
  - 2578-5885
  issn:
  - 2578-5893
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: ' Global well-posedness and equicontinuity for modified Korteweg–de Vries equations
  in modulation spaces'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '22069'
abstract:
- lang: eng
  text: "For slowly-varying initial data, solutions to the Ablowitz–Ladik system have
    been proven to converge to solutions of the cubic Schrödinger equation. In this
    paper we show that in the continuum limit, solutions to the Ablowitz–Ladik system
    with H^1 initial data may also converge to solutions of the modified Korteweg–de
    Vries equation. To exhibit this new limiting behavior, it suffices that the initial
    data is supported near the inflection points of the dispersion relation associated
    with the Ablowitz–Ladik system.\r\n\r\nOur arguments employ harmonic analysis
    tools, Strichartz estimates, and the conservation of mass and energy. Correspondingly,
    they are applicable beyond the completely integrable models of greatest interest
    to us."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Zhimeng
  full_name: Ouyang, Zhimeng
  last_name: Ouyang
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: Lei
  full_name: Wu, Lei
  last_name: Wu
citation:
  ama: Killip R, Ouyang Z, Vişan M, Wu L. The modified Korteweg–de Vries limit of
    the Ablowitz–Ladik system. <i>Discrete and Continuous Dynamical Systems</i>. 2025;45(3):821-846.
    doi:<a href="https://doi.org/10.3934/dcds.2024114">10.3934/dcds.2024114</a>
  apa: Killip, R., Ouyang, Z., Vişan, M., &#38; Wu, L. (2025). The modified Korteweg–de
    Vries limit of the Ablowitz–Ladik system. <i>Discrete and Continuous Dynamical
    Systems</i>. American Institute of Mathematical Sciences. <a href="https://doi.org/10.3934/dcds.2024114">https://doi.org/10.3934/dcds.2024114</a>
  chicago: Killip, Rowan, Zhimeng Ouyang, Monica Vişan, and Lei Wu. “The Modified
    Korteweg–de Vries Limit of the Ablowitz–Ladik System.” <i>Discrete and Continuous
    Dynamical Systems</i>. American Institute of Mathematical Sciences, 2025. <a href="https://doi.org/10.3934/dcds.2024114">https://doi.org/10.3934/dcds.2024114</a>.
  ieee: R. Killip, Z. Ouyang, M. Vişan, and L. Wu, “The modified Korteweg–de Vries
    limit of the Ablowitz–Ladik system,” <i>Discrete and Continuous Dynamical Systems</i>,
    vol. 45, no. 3. American Institute of Mathematical Sciences, pp. 821–846, 2025.
  ista: Killip R, Ouyang Z, Vişan M, Wu L. 2025. The modified Korteweg–de Vries limit
    of the Ablowitz–Ladik system. Discrete and Continuous Dynamical Systems. 45(3),
    821–846.
  mla: Killip, Rowan, et al. “The Modified Korteweg–de Vries Limit of the Ablowitz–Ladik
    System.” <i>Discrete and Continuous Dynamical Systems</i>, vol. 45, no. 3, American
    Institute of Mathematical Sciences, 2025, pp. 821–46, doi:<a href="https://doi.org/10.3934/dcds.2024114">10.3934/dcds.2024114</a>.
  short: R. Killip, Z. Ouyang, M. Vişan, L. Wu, Discrete and Continuous Dynamical
    Systems 45 (2025) 821–846.
das_tickbox: '1'
date_created: 2026-06-19T08:20:28Z
date_published: 2025-03-01T00:00:00Z
date_updated: 2026-06-30T07:34:20Z
day: '01'
ddc:
- '500'
doi: 10.3934/dcds.2024114
extern: '1'
external_id:
  arxiv:
  - '2404.02366'
intvolume: '        45'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2404.02366
mathsc:
- 37J70
- 37K10
- 37K60
- 35Q53
- 35Q55
month: '03'
oa: 1
oa_version: Preprint
page: 821-846
publication: Discrete and Continuous Dynamical Systems
publication_identifier:
  eissn:
  - 1553-5231
  issn:
  - 1078-0947
publication_status: published
publisher: American Institute of Mathematical Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: The modified Korteweg–de Vries limit of the Ablowitz–Ladik system
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 45
year: '2025'
...
---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
PlanS_conform: '1'
_id: '22079'
abstract:
- lang: eng
  text: 'We prove that the cubic nonlinear Schrödinger equation (both focusing and
    defocusing) is globally well-posed in H^s(R) for any regularity s > −1/2. Well-posedness
    has long been known for s ≥ 0, see [55], but not previously for any s < 0. The
    scaling-critical value s = −1/2 is necessarily excluded here, since instantaneous
    norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion)
    well-posedness of the real- and complex-valued modified Korteweg–de Vries equations
    in H^s(R) for any s > −1/2. The best regularity achieved previously was s ≥ 1/4
    (see [15, 24, 33, 39]). To overcome the failure of uniform continuity of the data-to-solution
    map, we employ the method of commuting flows introduced in [37]. In stark contrast
    with our arguments in [37], an essential ingredient in this paper is the demonstration
    of a local smoothing effect for both equations. Despite the nonperturbative nature
    of the well-posedness, the gain of derivatives matches that of the underlying
    linear equation. To compensate for the local nature of the smoothing estimates,
    we also demonstrate tightness of orbits. The proofs of both local smoothing and
    tightness rely on our discovery of a new one-parameter family of coercive microscopic
    conservation laws that remain meaningful at this low regularity. '
article_number: e6
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Benjamin
  full_name: Harrop-Griffiths, Benjamin
  last_name: Harrop-Griffiths
- 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
citation:
  ama: Harrop-Griffiths B, Killip R, Vişan M. Sharp well-posedness for the cubic NLS
    and mKdV in H^s(R). <i>Forum of Mathematics, Pi</i>. 2024;12. doi:<a href="https://doi.org/10.1017/fmp.2024.4">10.1017/fmp.2024.4</a>
  apa: Harrop-Griffiths, B., Killip, R., &#38; Vişan, M. (2024). Sharp well-posedness
    for the cubic NLS and mKdV in H^s(R). <i>Forum of Mathematics, Pi</i>. Cambridge
    University Press. <a href="https://doi.org/10.1017/fmp.2024.4">https://doi.org/10.1017/fmp.2024.4</a>
  chicago: Harrop-Griffiths, Benjamin, Rowan Killip, and Monica Vişan. “Sharp Well-Posedness
    for the Cubic NLS and MKdV in H^s(R).” <i>Forum of Mathematics, Pi</i>. Cambridge
    University Press, 2024. <a href="https://doi.org/10.1017/fmp.2024.4">https://doi.org/10.1017/fmp.2024.4</a>.
  ieee: B. Harrop-Griffiths, R. Killip, and M. Vişan, “Sharp well-posedness for the
    cubic NLS and mKdV in H^s(R),” <i>Forum of Mathematics, Pi</i>, vol. 12. Cambridge
    University Press, 2024.
  ista: Harrop-Griffiths B, Killip R, Vişan M. 2024. Sharp well-posedness for the
    cubic NLS and mKdV in H^s(R). Forum of Mathematics, Pi. 12, e6.
  mla: Harrop-Griffiths, Benjamin, et al. “Sharp Well-Posedness for the Cubic NLS
    and MKdV in H^s(R).” <i>Forum of Mathematics, Pi</i>, vol. 12, e6, Cambridge University
    Press, 2024, doi:<a href="https://doi.org/10.1017/fmp.2024.4">10.1017/fmp.2024.4</a>.
  short: B. Harrop-Griffiths, R. Killip, M. Vişan, Forum of Mathematics, Pi 12 (2024).
das_tickbox: '1'
date_created: 2026-06-19T08:26:10Z
date_published: 2024-04-02T00:00:00Z
date_updated: 2026-06-30T12:16:50Z
day: '02'
ddc:
- '500'
doi: 10.1017/fmp.2024.4
extern: '1'
external_id:
  arxiv:
  - '2003.05011'
has_accepted_license: '1'
intvolume: '        12'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1017/fmp.2024.4
mathsc:
- 35Q55
- 35Q53
- 37K10
month: '04'
oa: 1
oa_version: Published Version
publication: Forum of Mathematics, Pi
publication_identifier:
  eissn:
  - 2050-5086
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sharp well-posedness for the cubic NLS and mKdV in H^s(R)
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 12
year: '2024'
...
---
OA_place: repository
OA_type: green
_id: '22046'
abstract:
- lang: eng
  text: We show that solutions to the Ablowitz–Ladik system converge to solutions
    of the cubic nonlinear Schrödinger equation for merely L2 initial data. Furthermore,
    we consider initial data for this lattice model that excites Fourier modes near
    both critical points of the discrete dispersion relation and demonstrate convergence
    to a decoupled system of nonlinear Schrödinger equations.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Zhimeng
  full_name: Ouyang, Zhimeng
  last_name: Ouyang
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: Lei
  full_name: Wu, Lei
  last_name: Wu
citation:
  ama: Killip R, Ouyang Z, Vişan M, Wu L. Continuum limit for the Ablowitz–Ladik system.
    <i>Nonlinearity</i>. 2023;36(7):3751-3775. doi:<a href="https://doi.org/10.1088/1361-6544/acd978">10.1088/1361-6544/acd978</a>
  apa: Killip, R., Ouyang, Z., Vişan, M., &#38; Wu, L. (2023). Continuum limit for
    the Ablowitz–Ladik system. <i>Nonlinearity</i>. IOP Publishing. <a href="https://doi.org/10.1088/1361-6544/acd978">https://doi.org/10.1088/1361-6544/acd978</a>
  chicago: Killip, Rowan, Zhimeng Ouyang, Monica Vişan, and Lei Wu. “Continuum Limit
    for the Ablowitz–Ladik System.” <i>Nonlinearity</i>. IOP Publishing, 2023. <a
    href="https://doi.org/10.1088/1361-6544/acd978">https://doi.org/10.1088/1361-6544/acd978</a>.
  ieee: R. Killip, Z. Ouyang, M. Vişan, and L. Wu, “Continuum limit for the Ablowitz–Ladik
    system,” <i>Nonlinearity</i>, vol. 36, no. 7. IOP Publishing, pp. 3751–3775, 2023.
  ista: Killip R, Ouyang Z, Vişan M, Wu L. 2023. Continuum limit for the Ablowitz–Ladik
    system. Nonlinearity. 36(7), 3751–3775.
  mla: Killip, Rowan, et al. “Continuum Limit for the Ablowitz–Ladik System.” <i>Nonlinearity</i>,
    vol. 36, no. 7, IOP Publishing, 2023, pp. 3751–75, doi:<a href="https://doi.org/10.1088/1361-6544/acd978">10.1088/1361-6544/acd978</a>.
  short: R. Killip, Z. Ouyang, M. Vişan, L. Wu, Nonlinearity 36 (2023) 3751–3775.
das_tickbox: '1'
date_created: 2026-06-19T07:49:24Z
date_published: 2023-06-09T00:00:00Z
date_updated: 2026-06-25T07:54:44Z
day: '09'
doi: 10.1088/1361-6544/acd978
extern: '1'
external_id:
  arxiv:
  - '2206.02720'
intvolume: '        36'
issue: '7'
keyword:
- Ablowitz–Ladik
- continuum limit
- cubic NLS
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2206.02720
mathsc:
- 35Q55
- 37K05
- 37K10
month: '06'
oa: 1
oa_version: Preprint
page: 3751-3775
publication: Nonlinearity
publication_identifier:
  eissn:
  - 1361-6544
  issn:
  - 0951-7715
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: Continuum limit for the Ablowitz–Ladik system
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 36
year: '2023'
...