Scattering for the L2supercritical point NLS

Riccardo Adami; Reika Fukuizumi; Justin Holmer

doi:10.1090/tran/8065

Scattering for the L²supercritical point NLS

Riccardo Adami, Reika Fukuizumi, Justin Holmer

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. "Point"means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blowup profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ±∞ in the L2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

Original language	English
Pages (from-to)	35-60
Number of pages	26
Journal	Transactions of the American Mathematical Society
Volume	374
Issue number	1
DOIs	https://doi.org/10.1090/tran/8065
Publication status	Published - 2020
Externally published	Yes

Keywords

Nonlinear point interaction
Scattering
Schrödinger equation

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

Access to Document

10.1090/tran/8065

Cite this

@article{ce75e0e906f34da09082a05eca4a8059,

title = "Scattering for the L2supercritical point NLS",

abstract = "We consider the 1D nonlinear Schr{\"o}dinger equation with focusing point nonlinearity. {"}Point{"}means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blowup profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ±∞ in the L2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation. ",

keywords = "Nonlinear point interaction, Scattering, Schr{\"o}dinger equation",

author = "Riccardo Adami and Reika Fukuizumi and Justin Holmer",

note = "Funding Information: Received by the editors May 9, 2019, and, in revised form, October 26, 2019. 2010 Mathematics Subject Classification. Primary 35B40, 35Q55. Key words and phrases. Schr{\"o}dinger equation, nonlinear point interaction, scattering. This work was supported by JSPS KAKENHI Grant Number 15K04944. Publisher Copyright: {\textcopyright} 2020 by the authors.",

year = "2020",

doi = "10.1090/tran/8065",

language = "English",

volume = "374",

pages = "35--60",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

number = "1",

}

TY - JOUR

T1 - Scattering for the L2supercritical point NLS

AU - Adami, Riccardo

AU - Fukuizumi, Reika

AU - Holmer, Justin

N1 - Funding Information: Received by the editors May 9, 2019, and, in revised form, October 26, 2019. 2010 Mathematics Subject Classification. Primary 35B40, 35Q55. Key words and phrases. Schrödinger equation, nonlinear point interaction, scattering. This work was supported by JSPS KAKENHI Grant Number 15K04944. Publisher Copyright: © 2020 by the authors.

PY - 2020

Y1 - 2020

N2 - We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. "Point"means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blowup profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ±∞ in the L2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

AB - We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. "Point"means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blowup profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ±∞ in the L2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

KW - Nonlinear point interaction

KW - Scattering

KW - Schrödinger equation

UR - http://www.scopus.com/inward/record.url?scp=85097889468&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85097889468&partnerID=8YFLogxK

U2 - 10.1090/tran/8065

DO - 10.1090/tran/8065

M3 - Article

AN - SCOPUS:85097889468

VL - 374

SP - 35

EP - 60

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -