On the scattering problem for the nonlinear Schrödinger equation with a potential in 2D

Research output: Contribution to journalArticle

Abstract

We consider the scattering problem for the nonlinear Schrödinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator A(s) appearing in commutator relations. The equivalence between the operators −Δ V s/2 and −Δ s/2 in L 2 norm sense for 0≤s<1 is investigated by using free resolvent estimates and Gaussian estimates for the heat kernel of the Schrödinger operator −Δ V . Our main result guarantees the global existence of solutions and time decay of the solutions assuming initial data have small weighted Sobolev norms. Moreover, the global solutions obtained in the main result scatter.

Original languageEnglish
JournalPhysica D: Nonlinear Phenomena
DOIs
Publication statusPublished - 2019 Jan 1

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Electric commutators
Scattering Problems
Nonlinear equations
Resolvent Estimates
nonlinear equations
Nonlinear Equations
Scattering
estimates
Operator
scattering
norms
operators
Gaussian Estimates
Norm
Heat Kernel
Scatter
Commutator
Global Solution
Global Existence
Existence of Solutions

Keywords

  • Nonlinear Schrödinger equation
  • Resolvent estimates
  • Scattering problem
  • Strichartz estimates
  • Time decay estimates

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

Cite this

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title = "On the scattering problem for the nonlinear Schr{\"o}dinger equation with a potential in 2D",
abstract = "We consider the scattering problem for the nonlinear Schr{\"o}dinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator A(s) appearing in commutator relations. The equivalence between the operators −Δ V s/2 and −Δ s/2 in L 2 norm sense for 0≤s<1 is investigated by using free resolvent estimates and Gaussian estimates for the heat kernel of the Schr{\"o}dinger operator −Δ V . Our main result guarantees the global existence of solutions and time decay of the solutions assuming initial data have small weighted Sobolev norms. Moreover, the global solutions obtained in the main result scatter.",
keywords = "Nonlinear Schr{\"o}dinger equation, Resolvent estimates, Scattering problem, Strichartz estimates, Time decay estimates",
author = "Gueorguiev, {Vladimir Simeonov} and Chunhua Li",
year = "2019",
month = "1",
day = "1",
doi = "10.1016/j.physd.2019.03.010",
language = "English",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",

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T1 - On the scattering problem for the nonlinear Schrödinger equation with a potential in 2D

AU - Gueorguiev, Vladimir Simeonov

AU - Li, Chunhua

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We consider the scattering problem for the nonlinear Schrödinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator A(s) appearing in commutator relations. The equivalence between the operators −Δ V s/2 and −Δ s/2 in L 2 norm sense for 0≤s<1 is investigated by using free resolvent estimates and Gaussian estimates for the heat kernel of the Schrödinger operator −Δ V . Our main result guarantees the global existence of solutions and time decay of the solutions assuming initial data have small weighted Sobolev norms. Moreover, the global solutions obtained in the main result scatter.

AB - We consider the scattering problem for the nonlinear Schrödinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator A(s) appearing in commutator relations. The equivalence between the operators −Δ V s/2 and −Δ s/2 in L 2 norm sense for 0≤s<1 is investigated by using free resolvent estimates and Gaussian estimates for the heat kernel of the Schrödinger operator −Δ V . Our main result guarantees the global existence of solutions and time decay of the solutions assuming initial data have small weighted Sobolev norms. Moreover, the global solutions obtained in the main result scatter.

KW - Nonlinear Schrödinger equation

KW - Resolvent estimates

KW - Scattering problem

KW - Strichartz estimates

KW - Time decay estimates

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