PARTIAL DATA INVERSE PROBLEMS FOR THE NONLINEAR TIME-DEPENDENT SCHRÖDINGER EQUATION

Ru Yu Lai; Xuezhu Lu; Ting Zhou

doi:10.1137/23M1587993

PARTIAL DATA INVERSE PROBLEMS FOR THE NONLINEAR TIME-DEPENDENT SCHRÖDINGER EQUATION

Ru Yu Lai
, Xuezhu Lu
, Ting Zhou

School of Mathematics

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

2 Scopus citations

Abstract

In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient β(t, x) in the Schr\" odinger equation (iƌ_t + Δ + q(t, x))u + βu² = 0, from the boundary Dirichlet-to-Neumann (DN) map. In particular, we are interested in the partial data problem, in which the DN map is measured on a proper subset of the boundary. We show two results: a local uniqueness of the coefficient at the points where certain types of geometric optics solutions can reach, and a stability estimate based on the unique continuation property for the linear equation.

Original language	English (US)
Pages (from-to)	4712-4741
Number of pages	30
Journal	SIAM Journal on Mathematical Analysis
Volume	56
Issue number	4
DOIs	https://doi.org/10.1137/23M1587993
State	Published - Aug 2024

Bibliographical note

Publisher Copyright:
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Keywords

inverse problems
nonlinearity
time-dependent Schrödinger equation

Publisher link

10.1137/23M1587993

Find It

Find It at U of M Libraries

Cite this

@article{793e5b360a83406ebf1cdfb4bdf8cddd,

title = "PARTIAL DATA INVERSE PROBLEMS FOR THE NONLINEAR TIME-DEPENDENT SCHR{\"O}DINGER EQUATION",

abstract = "In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient β(t, x) in the Schr\textbackslash{}{"} odinger equation (iƌt + Δ + q(t, x))u + βu2 = 0, from the boundary Dirichlet-to-Neumann (DN) map. In particular, we are interested in the partial data problem, in which the DN map is measured on a proper subset of the boundary. We show two results: a local uniqueness of the coefficient at the points where certain types of geometric optics solutions can reach, and a stability estimate based on the unique continuation property for the linear equation.",

keywords = "inverse problems, nonlinearity, time-dependent Schr{\"o}dinger equation",

author = "Lai, \{Ru Yu\} and Xuezhu Lu and Ting Zhou",

note = "Publisher Copyright: Copyright {\textcopyright} by SIAM. Unauthorized reproduction of this article is prohibited.",

year = "2024",

month = aug,

doi = "10.1137/23M1587993",

language = "English (US)",

volume = "56",

pages = "4712--4741",

journal = "SIAM Journal on Mathematical Analysis",

issn = "0036-1410",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "4",

}

TY - JOUR

T1 - PARTIAL DATA INVERSE PROBLEMS FOR THE NONLINEAR TIME-DEPENDENT SCHRÖDINGER EQUATION

AU - Lai, Ru Yu

AU - Lu, Xuezhu

AU - Zhou, Ting

PY - 2024/8

Y1 - 2024/8

N2 - In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient β(t, x) in the Schr\" odinger equation (iƌt + Δ + q(t, x))u + βu2 = 0, from the boundary Dirichlet-to-Neumann (DN) map. In particular, we are interested in the partial data problem, in which the DN map is measured on a proper subset of the boundary. We show two results: a local uniqueness of the coefficient at the points where certain types of geometric optics solutions can reach, and a stability estimate based on the unique continuation property for the linear equation.

AB - In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient β(t, x) in the Schr\" odinger equation (iƌt + Δ + q(t, x))u + βu2 = 0, from the boundary Dirichlet-to-Neumann (DN) map. In particular, we are interested in the partial data problem, in which the DN map is measured on a proper subset of the boundary. We show two results: a local uniqueness of the coefficient at the points where certain types of geometric optics solutions can reach, and a stability estimate based on the unique continuation property for the linear equation.

KW - inverse problems

KW - nonlinearity

KW - time-dependent Schrödinger equation

UR - https://www.scopus.com/pages/publications/85199715377

UR - https://www.scopus.com/pages/publications/85199715377#tab=citedBy

U2 - 10.1137/23M1587993

DO - 10.1137/23M1587993

M3 - Article

AN - SCOPUS:85199715377

SN - 0036-1410

VL - 56

SP - 4712

EP - 4741

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 4

ER -

PARTIAL DATA INVERSE PROBLEMS FOR THE NONLINEAR TIME-DEPENDENT SCHRÖDINGER EQUATION

Abstract

Bibliographical note

Keywords

Publisher link

Find It

Other files and links

Fingerprint

Cite this