Inverse problems for a quasilinear hyperbolic equation with multiple unknowns

Yan Jiang; Hongyu Liu; Tianhao Ni; Kai Zhang

doi:10.1016/j.jfa.2025.110986

Inverse problems for a quasilinear hyperbolic equation with multiple unknowns

Yan Jiang, Hongyu Liu^*, Tianhao Ni, Kai Zhang^*

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

Abstract

We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form c(x)⁻²∂_t²u = Δ_g(u + F(x, u)) + G(x, u) on a compact Riemannian manifold (M, g) with boundary. We show that if F(x, u) is monomial and G(x, u) is analytic in u, then F, G and c as well as the associated initial data can be uniquely determined and reconstructed by the corresponding hyperbolic DtN (Dirichlet-to-Neumann) map. Our work leverages the construction of proper Gaussian beam solutions for quasilinear hyperbolic PDEs as well as their intriguing applications in conjunction with light-ray transforms and stationary phase techniques for related inverse problems. The results obtained are also of practical importance in assorted applications with nonlinear waves. © 2025 Elsevier Inc.

Original language	English
Article number	110986
Journal	Journal of Functional Analysis
Volume	289
Issue number	7
Online published	3 Apr 2025
DOIs	https://doi.org/10.1016/j.jfa.2025.110986
Publication status	Published - 1 Oct 2025

Funding

The work of Y. Jiang is supported in part by the National Key R&D Program of China (2024YFA1012302) and China Natural National Science Foundation (No. 123B2017). The work of H. Liu is supported by the Hong Kong RGC General Research Funds (Projects 11311122, 11300821, and 11304224), NSF/RGC Joint Research Fund (Project N_CityU101/21) and the ANR/RGC Joint Research Fund (Project A_CityU203/19). The work of K. Zhang is supported in part by China Natural National Science Foundation (Grant No. 12271207), and by the Fundamental Research Funds for the Central Universities, China.

Research Keywords

Gaussian beams
Inverse problems
Light-ray transform
Quasilinear hyperbolic equation

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RGC-funded

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title = "Inverse problems for a quasilinear hyperbolic equation with multiple unknowns",

abstract = "We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form c(x)−2∂t2u = Δg(u + F(x, u)) + G(x, u) on a compact Riemannian manifold (M, g) with boundary. We show that if F(x, u) is monomial and G(x, u) is analytic in u, then F, G and c as well as the associated initial data can be uniquely determined and reconstructed by the corresponding hyperbolic DtN (Dirichlet-to-Neumann) map. Our work leverages the construction of proper Gaussian beam solutions for quasilinear hyperbolic PDEs as well as their intriguing applications in conjunction with light-ray transforms and stationary phase techniques for related inverse problems. The results obtained are also of practical importance in assorted applications with nonlinear waves. {\textcopyright} 2025 Elsevier Inc.",

keywords = "Gaussian beams, Inverse problems, Light-ray transform, Quasilinear hyperbolic equation",

author = "Yan Jiang and Hongyu Liu and Tianhao Ni and Kai Zhang",

year = "2025",

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day = "1",

doi = "10.1016/j.jfa.2025.110986",

language = "English",

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journal = "Journal of Functional Analysis",

issn = "0022-1236",

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TY - JOUR

T1 - Inverse problems for a quasilinear hyperbolic equation with multiple unknowns

AU - Jiang, Yan

AU - Liu, Hongyu

AU - Ni, Tianhao

AU - Zhang, Kai

PY - 2025/10/1

Y1 - 2025/10/1

N2 - We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form c(x)−2∂t2u = Δg(u + F(x, u)) + G(x, u) on a compact Riemannian manifold (M, g) with boundary. We show that if F(x, u) is monomial and G(x, u) is analytic in u, then F, G and c as well as the associated initial data can be uniquely determined and reconstructed by the corresponding hyperbolic DtN (Dirichlet-to-Neumann) map. Our work leverages the construction of proper Gaussian beam solutions for quasilinear hyperbolic PDEs as well as their intriguing applications in conjunction with light-ray transforms and stationary phase techniques for related inverse problems. The results obtained are also of practical importance in assorted applications with nonlinear waves. © 2025 Elsevier Inc.

AB - We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form c(x)−2∂t2u = Δg(u + F(x, u)) + G(x, u) on a compact Riemannian manifold (M, g) with boundary. We show that if F(x, u) is monomial and G(x, u) is analytic in u, then F, G and c as well as the associated initial data can be uniquely determined and reconstructed by the corresponding hyperbolic DtN (Dirichlet-to-Neumann) map. Our work leverages the construction of proper Gaussian beam solutions for quasilinear hyperbolic PDEs as well as their intriguing applications in conjunction with light-ray transforms and stationary phase techniques for related inverse problems. The results obtained are also of practical importance in assorted applications with nonlinear waves. © 2025 Elsevier Inc.

KW - Gaussian beams

KW - Inverse problems

KW - Light-ray transform

KW - Quasilinear hyperbolic equation

UR - http://www.scopus.com/inward/record.url?scp=105002014940&partnerID=8YFLogxK

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-105002014940&origin=recordpage

U2 - 10.1016/j.jfa.2025.110986

DO - 10.1016/j.jfa.2025.110986

M3 - RGC 21 - Publication in refereed journal

SN - 0022-1236

VL - 289

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 7

M1 - 110986

ER -

Inverse problems for a quasilinear hyperbolic equation with multiple unknowns

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