---
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: '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'
...
---
OA_place: repository
OA_type: green
_id: '22042'
abstract:
- lang: eng
  text: "We study the L p-theory for the Schrödinger operatorLa with inverse-square
    potential\r\na|x|^−2. Our main result describes when L p-based Sobolev spaces
    defined in terms of the\r\noperator (La)^s/2 agree with those defined via (−\x02)^s/2.We
    consider all regularities 0 < s < 2.\r\nIn order to make the paper self-contained,
    we also review (with proofs) multiplier theorems,\r\nLittlewood–Paley theory,
    and Hardy-type inequalities associated to the operator La."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: R.
  full_name: Killip, R.
  last_name: Killip
- first_name: C.
  full_name: Miao, C.
  last_name: Miao
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: J.
  full_name: Zhang, J.
  last_name: Zhang
- first_name: J.
  full_name: Zheng, J.
  last_name: Zheng
citation:
  ama: Killip R, Miao C, Vişan M, Zhang J, Zheng J. Sobolev spaces adapted to the
    Schrödinger operator with inverse-square potential. <i>Mathematische Zeitschrift</i>.
    2018;288(3-4):1273-1298. doi:<a href="https://doi.org/10.1007/s00209-017-1934-8">10.1007/s00209-017-1934-8</a>
  apa: Killip, R., Miao, C., Vişan, M., Zhang, J., &#38; Zheng, J. (2018). Sobolev
    spaces adapted to the Schrödinger operator with inverse-square potential. <i>Mathematische
    Zeitschrift</i>. Springer Nature. <a href="https://doi.org/10.1007/s00209-017-1934-8">https://doi.org/10.1007/s00209-017-1934-8</a>
  chicago: Killip, R., C. Miao, Monica Vişan, J. Zhang, and J. Zheng. “Sobolev Spaces
    Adapted to the Schrödinger Operator with Inverse-Square Potential.” <i>Mathematische
    Zeitschrift</i>. Springer Nature, 2018. <a href="https://doi.org/10.1007/s00209-017-1934-8">https://doi.org/10.1007/s00209-017-1934-8</a>.
  ieee: R. Killip, C. Miao, M. Vişan, J. Zhang, and J. Zheng, “Sobolev spaces adapted
    to the Schrödinger operator with inverse-square potential,” <i>Mathematische Zeitschrift</i>,
    vol. 288, no. 3–4. Springer Nature, pp. 1273–1298, 2018.
  ista: Killip R, Miao C, Vişan M, Zhang J, Zheng J. 2018. Sobolev spaces adapted
    to the Schrödinger operator with inverse-square potential. Mathematische Zeitschrift.
    288(3–4), 1273–1298.
  mla: Killip, R., et al. “Sobolev Spaces Adapted to the Schrödinger Operator with
    Inverse-Square Potential.” <i>Mathematische Zeitschrift</i>, vol. 288, no. 3–4,
    Springer Nature, 2018, pp. 1273–98, doi:<a href="https://doi.org/10.1007/s00209-017-1934-8">10.1007/s00209-017-1934-8</a>.
  short: R. Killip, C. Miao, M. Vişan, J. Zhang, J. Zheng, Mathematische Zeitschrift
    288 (2018) 1273–1298.
date_created: 2026-06-19T07:46:14Z
date_published: 2018-04-01T00:00:00Z
date_updated: 2026-06-25T07:36:26Z
day: '01'
doi: 10.1007/s00209-017-1934-8
extern: '1'
external_id:
  arxiv:
  - '1503.02716'
intvolume: '       288'
issue: 3-4
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1503.02716
mathsc:
- 35P25
- 35Q55
month: '04'
oa: 1
oa_version: Preprint
page: 1273-1298
publication: Mathematische Zeitschrift
publication_identifier:
  eissn:
  - 1432-1823
  issn:
  - 0025-5874
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sobolev spaces adapted to the Schrödinger operator with inverse-square potential
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 288
year: '2018'
...
---
OA_place: repository
OA_type: green
_id: '22045'
abstract:
- lang: eng
  text: "We consider the initial-value problem for the cubic-quintic nonlinear Schrödinger
    equation (\U0001D456\U0001D715\U0001D461+Δ)⁢\U0001D713 =\U0001D6FC1⁢\U0001D713
    −\U0001D6FC3⁢|\U0001D713|2⁢\U0001D713 +\U0001D6FC5⁢|\U0001D713|4⁢\U0001D713 in
    three spatial dimensions in the class of solutions with |\U0001D713⁡(\U0001D465)|
    →\U0001D450 >0 as |\U0001D465| →∞. Here \U0001D6FC1, \U0001D6FC3, \U0001D6FC5,
    and \U0001D450 are such that \U0001D713⁡(\U0001D465) ≡\U0001D450 is an energetically
    stable equilibrium solution to this equation. Normalizing the boundary condition
    to \U0001D713⁡(\U0001D465) →1 as |\U0001D465| →∞, we study the associated initial-value
    problem for \U0001D462 =\U0001D713 −1 and prove a scattering result for small
    initial data in a weighted Sobolev space."
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 initial-value problem for the cubic-quintic
    NLS with nonvanishing boundary conditions. <i>SIAM Journal on Mathematical Analysis</i>.
    2018;50(3):2681-2739. doi:<a href="https://doi.org/10.1137/17m1116702">10.1137/17m1116702</a>
  apa: Killip, R., Murphy, J., &#38; Vişan, M. (2018). The initial-value problem for
    the cubic-quintic NLS with nonvanishing boundary conditions. <i>SIAM Journal on
    Mathematical Analysis</i>. Society for Industrial &#38; Applied Mathematics. <a
    href="https://doi.org/10.1137/17m1116702">https://doi.org/10.1137/17m1116702</a>
  chicago: Killip, Rowan, Jason Murphy, and Monica Vişan. “The Initial-Value Problem
    for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions.” <i>SIAM Journal
    on Mathematical Analysis</i>. Society for Industrial &#38; Applied Mathematics,
    2018. <a href="https://doi.org/10.1137/17m1116702">https://doi.org/10.1137/17m1116702</a>.
  ieee: R. Killip, J. Murphy, and M. Vişan, “The initial-value problem for the cubic-quintic
    NLS with nonvanishing boundary conditions,” <i>SIAM Journal on Mathematical Analysis</i>,
    vol. 50, no. 3. Society for Industrial &#38; Applied Mathematics, pp. 2681–2739,
    2018.
  ista: Killip R, Murphy J, Vişan M. 2018. The initial-value problem for the cubic-quintic
    NLS with nonvanishing boundary conditions. SIAM Journal on Mathematical Analysis.
    50(3), 2681–2739.
  mla: Killip, Rowan, et al. “The Initial-Value Problem for the Cubic-Quintic NLS
    with Nonvanishing Boundary Conditions.” <i>SIAM Journal on Mathematical Analysis</i>,
    vol. 50, no. 3, Society for Industrial &#38; Applied Mathematics, 2018, pp. 2681–739,
    doi:<a href="https://doi.org/10.1137/17m1116702">10.1137/17m1116702</a>.
  short: R. Killip, J. Murphy, M. Vişan, SIAM Journal on Mathematical Analysis 50
    (2018) 2681–2739.
das_tickbox: '1'
date_created: 2026-06-19T07:49:03Z
date_published: 2018-01-01T00:00:00Z
date_updated: 2026-06-25T07:49:21Z
day: '01'
doi: 10.1137/17m1116702
extern: '1'
external_id:
  arxiv:
  - '1702.04413'
intvolume: '        50'
issue: '3'
keyword:
- cubic-quintic NLS
- nonvanishing boundary conditions
- space-time resonances
- scattering
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1702.04413
mathsc:
- 35Q55
month: '01'
oa: 1
oa_version: Preprint
page: 2681-2739
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
  issn:
  - 0036-1410
  - 1095-7154
publication_status: published
publisher: Society for Industrial & Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: The initial-value problem for the cubic-quintic NLS with nonvanishing boundary
  conditions
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 50
year: '2018'
...
---
OA_place: repository
OA_type: green
_id: '22051'
abstract:
- lang: eng
  text: "We construct solutions with prescribed scattering state to the cubic-quintic
    NLS (mathematical formular)in three spatial dimensions in the class of solutions
    with (mathematical formular). This models disturbances in an infinite expanse
    of (quantum) fluid in its quiescent state— the limiting modulus c corresponds
    to a local minimum in the energy density.\r\nOur arguments build on work of Gustafson,
    Nakanishi, and Tsai on the (defocusing) Gross–Pitaevskii equation. The presence
    of an energy-critical nonlinearity and changes in the geometry of the energy\r\nfunctional
    add several new complexities. One new ingredient in our argument is a demonstration
    that\r\nsolutions of such (perturbed) energy-critical equations exhibit continuous
    dependence on the initial data\r\nwith respect to the weak topology on H1/x."
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 final-state problem for the cubic-quintic
    NLS with nonvanishing boundary conditions. <i>Analysis &#38; PDE</i>. 2016;9(7):1523-1574.
    doi:<a href="https://doi.org/10.2140/apde.2016.9.1523">10.2140/apde.2016.9.1523</a>
  apa: Killip, R., Murphy, J., &#38; Vişan, M. (2016). The final-state problem for
    the cubic-quintic NLS with nonvanishing boundary conditions. <i>Analysis &#38;
    PDE</i>. Mathematical Sciences Publishers. <a href="https://doi.org/10.2140/apde.2016.9.1523">https://doi.org/10.2140/apde.2016.9.1523</a>
  chicago: Killip, Rowan, Jason Murphy, and Monica Vişan. “The Final-State Problem
    for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions.” <i>Analysis
    &#38; PDE</i>. Mathematical Sciences Publishers, 2016. <a href="https://doi.org/10.2140/apde.2016.9.1523">https://doi.org/10.2140/apde.2016.9.1523</a>.
  ieee: R. Killip, J. Murphy, and M. Vişan, “The final-state problem for the cubic-quintic
    NLS with nonvanishing boundary conditions,” <i>Analysis &#38; PDE</i>, vol. 9,
    no. 7. Mathematical Sciences Publishers, pp. 1523–1574, 2016.
  ista: Killip R, Murphy J, Vişan M. 2016. The final-state problem for the cubic-quintic
    NLS with nonvanishing boundary conditions. Analysis &#38; PDE. 9(7), 1523–1574.
  mla: Killip, Rowan, et al. “The Final-State Problem for the Cubic-Quintic NLS with
    Nonvanishing Boundary Conditions.” <i>Analysis &#38; PDE</i>, vol. 9, no. 7, Mathematical
    Sciences Publishers, 2016, pp. 1523–74, doi:<a href="https://doi.org/10.2140/apde.2016.9.1523">10.2140/apde.2016.9.1523</a>.
  short: R. Killip, J. Murphy, M. Vişan, Analysis &#38; PDE 9 (2016) 1523–1574.
das_tickbox: '1'
date_created: 2026-06-19T07:54:01Z
date_published: 2016-11-07T00:00:00Z
date_updated: 2026-06-25T08:23:10Z
day: '07'
doi: 10.2140/apde.2016.9.1523
extern: '1'
external_id:
  arxiv:
  - '1506.06151'
intvolume: '         9'
issue: '7'
keyword:
- final-state problem
- wave operators
- cubic-quintic NLS
- nonvanishing boundary conditions
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1506.06151
mathsc:
- 35Q55
month: '11'
oa: 1
oa_version: Preprint
page: 1523-1574
publication: Analysis & PDE
publication_identifier:
  eissn:
  - 1948-206X
  issn:
  - 2157-5045
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: The final-state problem for the cubic-quintic NLS with nonvanishing boundary
  conditions
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 9
year: '2016'
...
---
OA_place: repository
OA_type: green
_id: '22053'
abstract:
- lang: eng
  text: We consider the Gross–Pitaevskii equation on R^4 and the cubic-quintic nonlinear
    Schrödinger equation (NLS) on R^3 with non-vanishing boundary conditions at spatial
    infinity. By viewing these equations as perturbations to the energy-critical NLS,
    we prove that they are globally well-posed in their energy spaces. In particular,
    we prove unconditional uniqueness in the energy spaces for these equations.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Tadahiro
  full_name: Oh, Tadahiro
  last_name: Oh
- first_name: Oana
  full_name: Pocovnicu, Oana
  last_name: Pocovnicu
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  ama: Killip R, Oh T, Pocovnicu O, Vişan M. Global well-posedness of the Gross–Pitaevskii
    and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary
    conditions. <i>Mathematical Research Letters</i>. 2013;19(5):969-986. doi:<a href="https://doi.org/10.4310/mrl.2012.v19.n5.a1">10.4310/mrl.2012.v19.n5.a1</a>
  apa: Killip, R., Oh, T., Pocovnicu, O., &#38; Vişan, M. (2013). Global well-posedness
    of the Gross–Pitaevskii and cubic-quintic nonlinear Schrödinger equations with
    non-vanishing boundary conditions. <i>Mathematical Research Letters</i>. International
    Press of Boston. <a href="https://doi.org/10.4310/mrl.2012.v19.n5.a1">https://doi.org/10.4310/mrl.2012.v19.n5.a1</a>
  chicago: Killip, Rowan, Tadahiro Oh, Oana Pocovnicu, and Monica Vişan. “Global Well-Posedness
    of the Gross–Pitaevskii and Cubic-Quintic Nonlinear Schrödinger Equations with
    Non-Vanishing Boundary Conditions.” <i>Mathematical Research Letters</i>. International
    Press of Boston, 2013. <a href="https://doi.org/10.4310/mrl.2012.v19.n5.a1">https://doi.org/10.4310/mrl.2012.v19.n5.a1</a>.
  ieee: R. Killip, T. Oh, O. Pocovnicu, and M. Vişan, “Global well-posedness of the
    Gross–Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing
    boundary conditions,” <i>Mathematical Research Letters</i>, vol. 19, no. 5. International
    Press of Boston, pp. 969–986, 2013.
  ista: Killip R, Oh T, Pocovnicu O, Vişan M. 2013. Global well-posedness of the Gross–Pitaevskii
    and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary
    conditions. Mathematical Research Letters. 19(5), 969–986.
  mla: Killip, Rowan, et al. “Global Well-Posedness of the Gross–Pitaevskii and Cubic-Quintic
    Nonlinear Schrödinger Equations with Non-Vanishing Boundary Conditions.” <i>Mathematical
    Research Letters</i>, vol. 19, no. 5, International Press of Boston, 2013, pp.
    969–86, doi:<a href="https://doi.org/10.4310/mrl.2012.v19.n5.a1">10.4310/mrl.2012.v19.n5.a1</a>.
  short: R. Killip, T. Oh, O. Pocovnicu, M. Vişan, Mathematical Research Letters 19
    (2013) 969–986.
das_tickbox: '1'
date_created: 2026-06-19T07:54:49Z
date_published: 2013-03-15T00:00:00Z
date_updated: 2026-06-25T08:33:18Z
day: '15'
doi: 10.4310/mrl.2012.v19.n5.a1
extern: '1'
external_id:
  arxiv:
  - '1112.1354'
intvolume: '        19'
issue: '5'
keyword:
- NLS
- Gross–Pitaevskii equation
- non-vanishing boundary condition
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1112.1354
mathsc:
- 35Q55
month: '03'
oa: 1
oa_version: Preprint
page: 969-986
publication: Mathematical Research Letters
publication_identifier:
  eissn:
  - 1945-001X
  issn:
  - 1073-2780
publication_status: published
publisher: International Press of Boston
quality_controlled: '1'
scopus_import: '1'
status: public
title: Global well-posedness of the Gross–Pitaevskii and cubic-quintic nonlinear Schrödinger
  equations with non-vanishing boundary conditions
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 19
year: '2013'
...
---
OA_place: repository
OA_type: green
_id: '22049'
abstract:
- lang: eng
  text: 'We consider the minimal mass m0 required for solutions to the mass-critical
    nonlinear Schrödinger (NLS) equation iut + Δu = μ|u|^4/d u to blow up. If m0 is
    finite, we show that there exists a minimal-mass solution blowing up (in the sense
    of an infinite spacetime norm) in both time directions, whose orbit in  is compact
    after quotienting out by the symmetries of the equation. A similar result is obtained
    for spherically symmetric solutions. Similar results were previously obtained
    by Keraani, [Keraani S.: On the blow-up phenomenon of the critical nonlinear Schrödinger
    equation. J. Funct. Anal. 235 (2006), 171–192], in dimensions 1, 2 and Begout
    and Vargas, [Begout P., Vargas A.: Mass concentration phenomena for the L2-critical
    nonlinear Schrödinger equation, preprint], in dimensions d ≥ 3 for the mass-critical
    NLS and by Kenig and Merle, [Kenig C., Merle F.: Global well-posedness, scattering,
    and blowup for the energy-critical, focusing, non-linear Schrödinger equation
    in the radial case, preprint], in the energy-critical case. In a subsequent paper
    we shall use this compactness result to establish global existence and scattering
    in  for the defocusing NLS in three and higher dimensions with spherically symmetric
    data.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Terence
  full_name: Tao, Terence
  last_name: Tao
- 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: Tao T, Vişan M, Zhang X. Minimal-mass blowup solutions of the mass-critical
    NLS. <i>Forum Mathematicum</i>. 2008;20(5):881-919. doi:<a href="https://doi.org/10.1515/forum.2008.042">10.1515/forum.2008.042</a>
  apa: Tao, T., Vişan, M., &#38; Zhang, X. (2008). Minimal-mass blowup solutions of
    the mass-critical NLS. <i>Forum Mathematicum</i>. De Gruyter. <a href="https://doi.org/10.1515/forum.2008.042">https://doi.org/10.1515/forum.2008.042</a>
  chicago: Tao, Terence, Monica Vişan, and Xiaoyi Zhang. “Minimal-Mass Blowup Solutions
    of the Mass-Critical NLS.” <i>Forum Mathematicum</i>. De Gruyter, 2008. <a href="https://doi.org/10.1515/forum.2008.042">https://doi.org/10.1515/forum.2008.042</a>.
  ieee: T. Tao, M. Vişan, and X. Zhang, “Minimal-mass blowup solutions of the mass-critical
    NLS,” <i>Forum Mathematicum</i>, vol. 20, no. 5. De Gruyter, pp. 881–919, 2008.
  ista: Tao T, Vişan M, Zhang X. 2008. Minimal-mass blowup solutions of the mass-critical
    NLS. Forum Mathematicum. 20(5), 881–919.
  mla: Tao, Terence, et al. “Minimal-Mass Blowup Solutions of the Mass-Critical NLS.”
    <i>Forum Mathematicum</i>, vol. 20, no. 5, De Gruyter, 2008, pp. 881–919, doi:<a
    href="https://doi.org/10.1515/forum.2008.042">10.1515/forum.2008.042</a>.
  short: T. Tao, M. Vişan, X. Zhang, Forum Mathematicum 20 (2008) 881–919.
das_tickbox: '1'
date_created: 2026-06-19T07:53:12Z
date_published: 2008-11-03T00:00:00Z
date_updated: 2026-06-25T08:15:22Z
day: '03'
doi: 10.1515/forum.2008.042
extern: '1'
external_id:
  arxiv:
  - math/0609690
intvolume: '        20'
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.math/0609690
mathsc:
- 35Q55
month: '11'
oa: 1
oa_version: Preprint
page: 881-919
publication: Forum Mathematicum
publication_identifier:
  eissn:
  - 1435-5337
  issn:
  - 0933-7741
publication_status: published
publisher: De Gruyter
quality_controlled: '1'
scopus_import: '1'
status: public
title: Minimal-mass blowup solutions of the mass-critical NLS
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 20
year: '2008'
...
---
OA_place: repository
OA_type: green
_id: '22047'
abstract:
- lang: eng
  text: "We undertake a comprehensive study of the nonlinear Schrödinger equation
    (mathematical formular) where u(t, x) is a complex-valued function in spacetime
    R, xRn/x, λ1 and λ2 are nonzero real constants, and (mathematical formular). We
    address questions related to local and global well-posedness, finite time blowup,
    and asymptotic behaviour. Scattering is considered both in the energy space H^1(ℝ
    n ) and in the pseudoconformal space Σ := {f ∈ H^1(ℝ^n); xf ∈ L^2(ℝ^n)}. Of particular
    interest is the case when both nonlinearities are defocusing and correspond to
    the L2/x-critical, respectively H1/x-critical NLS, that is, λ1, λ2 > 0 and (mathematical
    formular) . The results at the endpoint p1= 4/n are conditional on a conjectured
    global existence and spacetime estimate for the L2/x-critical nonlinear Schrödinger
    equation, which has been verified in dimensions n ≥ 2 for radial data in Tao et
    al. (Tao et al. to appear a,b) and Killip et al. (preprint).\r\nAs an off-shoot
    of our analysis, we also obtain a new, simpler proof of scattering in H1/x for
    solutions to the nonlinear Schrödinger equation (mathematical formular) with 4/n
    < p < 4/n-2, which was first obtained by Ginibre and Velo (Citation1985)."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Terence
  full_name: Tao, Terence
  last_name: Tao
- 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: Tao T, Vişan M, Zhang X. The nonlinear Schrödinger equation with combined power-type
    nonlinearities. <i>Communications in Partial Differential Equations</i>. 2007;32(8):1281-1343.
    doi:<a href="https://doi.org/10.1080/03605300701588805">10.1080/03605300701588805</a>
  apa: Tao, T., Vişan, M., &#38; Zhang, X. (2007). The nonlinear Schrödinger equation
    with combined power-type nonlinearities. <i>Communications in Partial Differential
    Equations</i>. Informa UK Limited. <a href="https://doi.org/10.1080/03605300701588805">https://doi.org/10.1080/03605300701588805</a>
  chicago: Tao, Terence, Monica Vişan, and Xiaoyi Zhang. “The Nonlinear Schrödinger
    Equation with Combined Power-Type Nonlinearities.” <i>Communications in Partial
    Differential Equations</i>. Informa UK Limited, 2007. <a href="https://doi.org/10.1080/03605300701588805">https://doi.org/10.1080/03605300701588805</a>.
  ieee: T. Tao, M. Vişan, and X. Zhang, “The nonlinear Schrödinger equation with combined
    power-type nonlinearities,” <i>Communications in Partial Differential Equations</i>,
    vol. 32, no. 8. Informa UK Limited, pp. 1281–1343, 2007.
  ista: Tao T, Vişan M, Zhang X. 2007. The nonlinear Schrödinger equation with combined
    power-type nonlinearities. Communications in Partial Differential Equations. 32(8),
    1281–1343.
  mla: Tao, Terence, et al. “The Nonlinear Schrödinger Equation with Combined Power-Type
    Nonlinearities.” <i>Communications in Partial Differential Equations</i>, vol.
    32, no. 8, Informa UK Limited, 2007, pp. 1281–343, doi:<a href="https://doi.org/10.1080/03605300701588805">10.1080/03605300701588805</a>.
  short: T. Tao, M. Vişan, X. Zhang, Communications in Partial Differential Equations
    32 (2007) 1281–1343.
das_tickbox: '1'
date_created: 2026-06-19T07:49:46Z
date_published: 2007-08-29T00:00:00Z
date_updated: 2026-06-25T08:04:20Z
day: '29'
doi: 10.1080/03605300701588805
extern: '1'
external_id:
  arxiv:
  - math/0511070
intvolume: '        32'
issue: '8'
keyword:
- Energy-critical
- Mass-critical
- Nonlinear Schrödinger equation
- Wellposedness
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.math/0511070
mathsc:
- 35Q55
month: '08'
oa: 1
oa_version: Preprint
page: 1281-1343
publication: Communications in Partial Differential Equations
publication_identifier:
  eissn:
  - 1532-4133
  issn:
  - 0360-5302
publication_status: published
publisher: Informa UK Limited
quality_controlled: '1'
scopus_import: '1'
status: public
title: The nonlinear Schrödinger equation with combined power-type nonlinearities
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 32
year: '2007'
...
---
OA_place: repository
OA_type: green
_id: '22050'
abstract:
- lang: eng
  text: "We obtain global well-posedness, scattering, and global L2(n+2)/(n−2)/t,x
    space-time\r\nbounds for energy-space solutions to the energy-critical nonlinear
    Schrodinger (NLS) ¨\r\nequation in Rt × Rn/x , n ≥ 5."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  ama: Vişan M. The defocusing energy-critical nonlinear Schrödinger equation in higher
    dimensions. <i>Duke Mathematical Journal</i>. 2007;138(2):281-374. doi:<a href="https://doi.org/10.1215/s0012-7094-07-13825-0">10.1215/s0012-7094-07-13825-0</a>
  apa: Vişan, M. (2007). The defocusing energy-critical nonlinear Schrödinger equation
    in higher dimensions. <i>Duke Mathematical Journal</i>. Duke University Press.
    <a href="https://doi.org/10.1215/s0012-7094-07-13825-0">https://doi.org/10.1215/s0012-7094-07-13825-0</a>
  chicago: Vişan, Monica. “The Defocusing Energy-Critical Nonlinear Schrödinger Equation
    in Higher Dimensions.” <i>Duke Mathematical Journal</i>. Duke University Press,
    2007. <a href="https://doi.org/10.1215/s0012-7094-07-13825-0">https://doi.org/10.1215/s0012-7094-07-13825-0</a>.
  ieee: M. Vişan, “The defocusing energy-critical nonlinear Schrödinger equation in
    higher dimensions,” <i>Duke Mathematical Journal</i>, vol. 138, no. 2. Duke University
    Press, pp. 281–374, 2007.
  ista: Vişan M. 2007. The defocusing energy-critical nonlinear Schrödinger equation
    in higher dimensions. Duke Mathematical Journal. 138(2), 281–374.
  mla: Vişan, Monica. “The Defocusing Energy-Critical Nonlinear Schrödinger Equation
    in Higher Dimensions.” <i>Duke Mathematical Journal</i>, vol. 138, no. 2, Duke
    University Press, 2007, pp. 281–374, doi:<a href="https://doi.org/10.1215/s0012-7094-07-13825-0">10.1215/s0012-7094-07-13825-0</a>.
  short: M. Vişan, Duke Mathematical Journal 138 (2007) 281–374.
das_tickbox: '1'
date_created: 2026-06-19T07:53:37Z
date_published: 2007-06-01T00:00:00Z
date_updated: 2026-06-25T08:18:44Z
day: '01'
doi: 10.1215/s0012-7094-07-13825-0
extern: '1'
external_id:
  arxiv:
  - math/0508298
intvolume: '       138'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.math/0508298
mathsc:
- 35Q55
month: '06'
oa: 1
oa_version: Preprint
page: 281-374
publication: Duke Mathematical Journal
publication_identifier:
  issn:
  - 0012-7094
publication_status: published
publisher: Duke University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 138
year: '2007'
...