---
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'
fulldoi: https://doi.org/10.1137/17m1116702
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'
fulldoi: https://doi.org/10.2140/apde.2016.9.1523
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'
...