---
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'
fulldoi: https://doi.org/10.3934/dcds.2024114
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'
...
---
OA_type: free access
_id: '17231'
abstract:
- lang: eng
  text: In the class of projective billiards, which contains the usual billiards,
    we exhibit counter-examples to Ivrii's conjecture, which states that in any planar
    billiard with smooth boundary the set of periodic orbits has zero measure. The
    counter-examples are polygons admitting a 2-parameters family of n-periodic orbits,
    with n being either 3 or any even integer greater than 4.
article_processing_charge: No
article_type: original
author:
- first_name: Corentin
  full_name: Fiorebe, Corentin
  id: 06619f18-9070-11eb-847d-d1ee780bd88b
  last_name: Fiorebe
citation:
  ama: Fiorebe C. Examples of projective billiards with open sets of periodic orbits.
    <i>Discrete and Continuous Dynamical Systems- Series A</i>. 2024;44(11):3287-3301.
    doi:<a href="https://doi.org/10.3934/dcds.2024059">10.3934/dcds.2024059</a>
  apa: Fiorebe, C. (2024). Examples of projective billiards with open sets of periodic
    orbits. <i>Discrete and Continuous Dynamical Systems- Series A</i>. AIMS. <a href="https://doi.org/10.3934/dcds.2024059">https://doi.org/10.3934/dcds.2024059</a>
  chicago: Fiorebe, Corentin. “Examples of Projective Billiards with Open Sets of
    Periodic Orbits.” <i>Discrete and Continuous Dynamical Systems- Series A</i>.
    AIMS, 2024. <a href="https://doi.org/10.3934/dcds.2024059">https://doi.org/10.3934/dcds.2024059</a>.
  ieee: C. Fiorebe, “Examples of projective billiards with open sets of periodic orbits,”
    <i>Discrete and Continuous Dynamical Systems- Series A</i>, vol. 44, no. 11. AIMS,
    pp. 3287–3301, 2024.
  ista: Fiorebe C. 2024. Examples of projective billiards with open sets of periodic
    orbits. Discrete and Continuous Dynamical Systems- Series A. 44(11), 3287–3301.
  mla: Fiorebe, Corentin. “Examples of Projective Billiards with Open Sets of Periodic
    Orbits.” <i>Discrete and Continuous Dynamical Systems- Series A</i>, vol. 44,
    no. 11, AIMS, 2024, pp. 3287–301, doi:<a href="https://doi.org/10.3934/dcds.2024059">10.3934/dcds.2024059</a>.
  short: C. Fiorebe, Discrete and Continuous Dynamical Systems- Series A 44 (2024)
    3287–3301.
corr_author: '1'
date_created: 2024-07-14T22:01:10Z
date_published: 2024-11-01T00:00:00Z
date_updated: 2026-08-12T06:19:26Z
day: '01'
ddc:
- '500'
department:
- _id: VaKa
doi: 10.3934/dcds.2024059
external_id:
  isi:
  - '001230091000001'
fulldoi: https://doi.org/10.3934/dcds.2024059
intvolume: '        44'
isi: 1
issue: '11'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.3934/dcds.2024059
month: '11'
oa: 1
oa_version: Published Version
page: 3287-3301
publication: Discrete and Continuous Dynamical Systems- Series A
publication_identifier:
  eissn:
  - 1553-5231
  issn:
  - 1078-0947
publication_status: published
publisher: AIMS
quality_controlled: '1'
scopus_import: '1'
status: public
title: Examples of projective billiards with open sets of periodic orbits
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 44
year: '2024'
...
---
OA_place: repository
OA_type: green
_id: '22030'
abstract:
- lang: eng
  text: We consider the mass-subcritical NLS in dimensions d>=3 with radial initial
    data. In the defocusing case, we prove that any solution that remains bounded
    in the critical Sobolev space throughout its lifespan must be global and scatter.
    In the focusing case, we prove the existence of a threshold solution that has
    a compact flow.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Satoshi
  full_name: Masaki, Satoshi
  last_name: Masaki
- 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, Masaki S, Murphy J, Vişan M. The radial mass-subcritical NLS in negative
    order Sobolev spaces. <i>Discrete and Continuous Dynamical Systems</i>. 2019;39(1):553-583.
    doi:<a href="https://doi.org/10.3934/dcds.2019023">10.3934/dcds.2019023</a>
  apa: Killip, R., Masaki, S., Murphy, J., &#38; Vişan, M. (2019). The radial mass-subcritical
    NLS in negative order Sobolev spaces. <i>Discrete and Continuous Dynamical Systems</i>.
    American Institute of Mathematical Sciences. <a href="https://doi.org/10.3934/dcds.2019023">https://doi.org/10.3934/dcds.2019023</a>
  chicago: Killip, Rowan, Satoshi Masaki, Jason Murphy, and Monica Vişan. “The Radial
    Mass-Subcritical NLS in Negative Order Sobolev Spaces.” <i>Discrete and Continuous
    Dynamical Systems</i>. American Institute of Mathematical Sciences, 2019. <a href="https://doi.org/10.3934/dcds.2019023">https://doi.org/10.3934/dcds.2019023</a>.
  ieee: R. Killip, S. Masaki, J. Murphy, and M. Vişan, “The radial mass-subcritical
    NLS in negative order Sobolev spaces,” <i>Discrete and Continuous Dynamical Systems</i>,
    vol. 39, no. 1. American Institute of Mathematical Sciences, pp. 553–583, 2019.
  ista: Killip R, Masaki S, Murphy J, Vişan M. 2019. The radial mass-subcritical NLS
    in negative order Sobolev spaces. Discrete and Continuous Dynamical Systems. 39(1),
    553–583.
  mla: Killip, Rowan, et al. “The Radial Mass-Subcritical NLS in Negative Order Sobolev
    Spaces.” <i>Discrete and Continuous Dynamical Systems</i>, vol. 39, no. 1, American
    Institute of Mathematical Sciences, 2019, pp. 553–83, doi:<a href="https://doi.org/10.3934/dcds.2019023">10.3934/dcds.2019023</a>.
  short: R. Killip, S. Masaki, J. Murphy, M. Vişan, Discrete and Continuous Dynamical
    Systems 39 (2019) 553–583.
das_tickbox: '1'
date_created: 2026-06-19T07:41:46Z
date_published: 2019-01-01T00:00:00Z
date_updated: 2026-06-22T11:06:54Z
day: '01'
doi: 10.3934/dcds.2019023
extern: '1'
external_id:
  arxiv:
  - '1804.06753'
fulldoi: https://doi.org/10.3934/dcds.2019023
intvolume: '        39'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1804.06753
month: '01'
oa: 1
oa_version: Preprint
page: 553-583
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 radial mass-subcritical NLS in negative order Sobolev spaces
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 39
year: '2019'
...
---
OA_place: repository
OA_type: green
_id: '22033'
abstract:
- lang: eng
  text: "We consider the defocusing energy-critical nonlinear Schrödinger\r\nequation
    with inverse-square potential iut = −∆u + a|x|^−2u + |u|^4u in three\r\nspace
    dimensions. We prove global well-posedness and scattering for a >− 1/4 + 1/25.
    We also carry out the variational analysis needed to treat the focusing case."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Changxing
  full_name: Miao, Changxing
  last_name: Miao
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: Junyong
  full_name: Zhang, Junyong
  last_name: Zhang
- first_name: Jiqiang
  full_name: Zheng, Jiqiang
  last_name: Zheng
citation:
  ama: Killip R, Miao C, Vişan M, Zhang J, Zheng J. The energy-critical NLS with inverse-square
    potential. <i>Discrete and Continuous Dynamical Systems</i>. 2017;37(7):3831-3866.
    doi:<a href="https://doi.org/10.3934/dcds.2017162">10.3934/dcds.2017162</a>
  apa: Killip, R., Miao, C., Vişan, M., Zhang, J., &#38; Zheng, J. (2017). The energy-critical
    NLS with inverse-square potential. <i>Discrete and Continuous Dynamical Systems</i>.
    American Institute of Mathematical Sciences. <a href="https://doi.org/10.3934/dcds.2017162">https://doi.org/10.3934/dcds.2017162</a>
  chicago: Killip, Rowan, Changxing Miao, Monica Vişan, Junyong Zhang, and Jiqiang
    Zheng. “The Energy-Critical NLS with Inverse-Square Potential.” <i>Discrete and
    Continuous Dynamical Systems</i>. American Institute of Mathematical Sciences,
    2017. <a href="https://doi.org/10.3934/dcds.2017162">https://doi.org/10.3934/dcds.2017162</a>.
  ieee: R. Killip, C. Miao, M. Vişan, J. Zhang, and J. Zheng, “The energy-critical
    NLS with inverse-square potential,” <i>Discrete and Continuous Dynamical Systems</i>,
    vol. 37, no. 7. American Institute of Mathematical Sciences, pp. 3831–3866, 2017.
  ista: Killip R, Miao C, Vişan M, Zhang J, Zheng J. 2017. The energy-critical NLS
    with inverse-square potential. Discrete and Continuous Dynamical Systems. 37(7),
    3831–3866.
  mla: Killip, Rowan, et al. “The Energy-Critical NLS with Inverse-Square Potential.”
    <i>Discrete and Continuous Dynamical Systems</i>, vol. 37, no. 7, American Institute
    of Mathematical Sciences, 2017, pp. 3831–66, doi:<a href="https://doi.org/10.3934/dcds.2017162">10.3934/dcds.2017162</a>.
  short: R. Killip, C. Miao, M. Vişan, J. Zhang, J. Zheng, Discrete and Continuous
    Dynamical Systems 37 (2017) 3831–3866.
das_tickbox: '1'
date_created: 2026-06-19T07:42:59Z
date_published: 2017-07-01T00:00:00Z
date_updated: 2026-06-22T12:47:32Z
day: '01'
doi: 10.3934/dcds.2017162
extern: '1'
external_id:
  arxiv:
  - '1509.05822'
fulldoi: https://doi.org/10.3934/dcds.2017162
intvolume: '        37'
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1509.05822
month: '07'
oa: 1
oa_version: Preprint
page: 3831-3866
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 energy-critical NLS with inverse-square potential
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 37
year: '2017'
...
---
OA_place: repository
OA_type: green
_id: '22056'
abstract:
- lang: eng
  text: We consider the mass-critical generalized Korteweg{de Vries equation (∂t +
    ∂xxx)u = ±∂ x(u 5) for real-valued functions u(t; x). We prove that if the global
    well-posedness and scattering conjecture for this equation failed, then, conditional
    on a positive answer to the global well-posedness and scattering conjecture for
    the masscritical nonlinear Schrffodinger equation (-i∂ t + ∂xx)u = ±(|u| 4u),
    there exists a minimal-mass blowup solution to the mass-critical generalized KdV
    equation which is almost periodic modulo the symmetries of the equation. Moreover,
    we can guarantee that this minimal-mass blowup solution is either a self-similar
    solution, a soliton-like solution, or a double high-to-low frequency cascade solution.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Soonsik
  full_name: Kwon, Soonsik
  last_name: Kwon
- first_name: Shuanglin
  full_name: Shao, Shuanglin
  last_name: Shao
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  ama: Killip R, Kwon S, Shao S, Vişan M. On the mass-critical generalized KdV equation.
    <i>Discrete and Continuous Dynamical Systems</i>. 2012;32(1):191-221. doi:<a href="https://doi.org/10.3934/dcds.2012.32.191">10.3934/dcds.2012.32.191</a>
  apa: Killip, R., Kwon, S., Shao, S., &#38; Vişan, M. (2012). On the mass-critical
    generalized KdV equation. <i>Discrete and Continuous Dynamical Systems</i>. American
    Institute of Mathematical Sciences. <a href="https://doi.org/10.3934/dcds.2012.32.191">https://doi.org/10.3934/dcds.2012.32.191</a>
  chicago: Killip, Rowan, Soonsik Kwon, Shuanglin Shao, and Monica Vişan. “On the
    Mass-Critical Generalized KdV Equation.” <i>Discrete and Continuous Dynamical
    Systems</i>. American Institute of Mathematical Sciences, 2012. <a href="https://doi.org/10.3934/dcds.2012.32.191">https://doi.org/10.3934/dcds.2012.32.191</a>.
  ieee: R. Killip, S. Kwon, S. Shao, and M. Vişan, “On the mass-critical generalized
    KdV equation,” <i>Discrete and Continuous Dynamical Systems</i>, vol. 32, no.
    1. American Institute of Mathematical Sciences, pp. 191–221, 2012.
  ista: Killip R, Kwon S, Shao S, Vişan M. 2012. On the mass-critical generalized
    KdV equation. Discrete and Continuous Dynamical Systems. 32(1), 191–221.
  mla: Killip, Rowan, et al. “On the Mass-Critical Generalized KdV Equation.” <i>Discrete
    and Continuous Dynamical Systems</i>, vol. 32, no. 1, American Institute of Mathematical
    Sciences, 2012, pp. 191–221, doi:<a href="https://doi.org/10.3934/dcds.2012.32.191">10.3934/dcds.2012.32.191</a>.
  short: R. Killip, S. Kwon, S. Shao, M. Vişan, Discrete and Continuous Dynamical
    Systems 32 (2012) 191–221.
das_tickbox: '1'
date_created: 2026-06-19T07:57:09Z
date_published: 2012-01-01T00:00:00Z
date_updated: 2026-06-29T09:59:46Z
day: '01'
doi: 10.3934/dcds.2012.32.191
extern: '1'
external_id:
  arxiv:
  - '0907.5412'
fulldoi: https://doi.org/10.3934/dcds.2012.32.191
intvolume: '        32'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.0907.5412
mathsc:
- 35Q53
month: '01'
oa: 1
oa_version: Preprint
page: 191-221
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: On the mass-critical generalized KdV equation
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 32
year: '2012'
...