Inverse problems for a quasilinear hyperbolic equation with multiple unknowns
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866913578699718656 |
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| author | Jiang, Yan Liu, Hongyu Ni, Tianhao Zhang, Kai |
| author_facet | Jiang, Yan Liu, Hongyu Ni, Tianhao Zhang, Kai |
| contents | We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form ${c(x)^{-2}}\partial_t^2u=Δ_g(u+F(x, u))+G(x, u)$ 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 of applications with nonlinear waves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09917 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inverse problems for a quasilinear hyperbolic equation with multiple unknowns Jiang, Yan Liu, Hongyu Ni, Tianhao Zhang, Kai Analysis of PDEs We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form ${c(x)^{-2}}\partial_t^2u=Δ_g(u+F(x, u))+G(x, u)$ 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 of applications with nonlinear waves. |
| title | Inverse problems for a quasilinear hyperbolic equation with multiple unknowns |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.09917 |