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

Vladimir Simeonov Gueorguiev; Chunhua Li

doi:10.1016/j.physd.2019.03.010

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

Vladimir Simeonov Gueorguiev, Chunhua Li

Graduate School of Advanced Science and Engineering

Research output: Contribution to journal › Article

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 language	English
Journal	Physica D: Nonlinear Phenomena
DOIs	https://doi.org/10.1016/j.physd.2019.03.010
Publication status	Published - 2019 Jan 1

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

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Gueorguiev, V. S., & Li, C. (2019). On the scattering problem for the nonlinear Schrödinger equation with a potential in 2D. Physica D: Nonlinear Phenomena. https://doi.org/10.1016/j.physd.2019.03.010

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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",

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day = "1",

doi = "10.1016/j.physd.2019.03.010",

language = "English",

journal = "Physica D: Nonlinear Phenomena",

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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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