Abstract
An inverse problem for electrodynamic equations is considered. It is assumed that the electric current depends nonlinearly of the electric tension. This dependence is determined by seven finite functions of space variables. A direct problem for electrodynamic equations with a running plane wave going in direction ν from infinity is stated.
Then traces of solutions of this direct problem on some bounded surface in
Funding statement: This work was carried out within the framework of the state assignment for Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, project no. FWNF-2022-0009.
References
[1] X. Chen, M. Lassas, L. Oksanen and G. Paternain, Detection of Hermitian connections in wave equations with cubic non-linearity, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 7, 2191–2232. 10.4171/jems/1136Suche in Google Scholar
[2] P. Hintz and G. Uhlmann, Reconstruction of Lorentzian manifolds from boundary light observation sets, Int. Math. Res. Not. IMRN 2019 (2019), no. 22, 6949–6987. 10.1093/imrn/rnx320Suche in Google Scholar
[3] P. Hintz, G. Uhlmann and J. Zhai, An inverse boundary value problem for a semilinear wave equation on Lorentzian manifolds, Int. Math. Res. Not. IMRN 2022 (2022), no. 17, 13181–13211. 10.1093/imrn/rnab088Suche in Google Scholar
[4] Y. Kurylev, M. Lassas and G. Uhlmann, Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Invent. Math. 212 (2018), no. 3, 781–857. 10.1007/s00222-017-0780-ySuche in Google Scholar
[5] M. Lassas, Inverse problems for linear and non-linear hyperbolic equations, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited Lectures, World Scientific, Hackensack (2018), 3751–3771. 10.1142/9789813272880_0199Suche in Google Scholar
[6] M. Lassas, T. Liimatainen, L. Potenciano-Machado and T. Tyni, Uniqueness, reconstruction and stability for an inverse problem of a semi-linear wave equation, J. Differential Equations 337 (2022), 395–435. 10.1016/j.jde.2022.08.010Suche in Google Scholar
[7] M. Lassas, G. Uhlmann and Y. Wang, Inverse problems for semilinear wave equations on Lorentzian manifolds, Comm. Math. Phys. 360 (2018), no. 2, 555–609. 10.1007/s00220-018-3135-7Suche in Google Scholar
[8] R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry, Sov. Math. Dokl. 18 (1977), no. 1, 27–31. Suche in Google Scholar
[9] V. G. Romanov, An inverse problem for a semilinear wave equation, Dokl. Math. 105 (2022), no. 3, 166–170. 10.1134/S1064562422030097Suche in Google Scholar
[10] V. G. Romanov, An inverse problem for the wave equation with nonlinear dumping, Sib. Math. J. 64 (2023), no. 3, 670–685. 10.1134/S003744662303014XSuche in Google Scholar
[11] V. G. Romanov, An inverse problem for the wave equation with two nonlinear terms, Differ. Equ. 60 (2024), no. 4, 479–491. 10.1134/S0012266124040074Suche in Google Scholar
[12] V. G. Romanov, A stability estimate for a solution to an inverse problem for a nonlinear hyperbolic equation, Sib. Math. J. 65 (2024), no. 3, 611–626. 10.1134/S0037446624030108Suche in Google Scholar
[13] V. G. Romanov, An inverse problem for the semilinear wave equation with a nonlinear integral operator, Sib. Math. J. 66 (2025), no. 2, 326–344. 10.1134/S0037446625020107Suche in Google Scholar
[14] V. G. Romanov and T. V. Bugueva, An inverse problem for a nonlinear hyperbolic equation, Eurasian J. Math. Comp. Appl. 12 (2024), no. 2, 134–154. 10.32523/2306-6172-2024-12-2-134-154Suche in Google Scholar
[15] V. G. Romanov and T. V. Bugueva, An one-dimensional inverse problem for the wave equation, Eurasian J. Math. Comp. Appl. 12 (2024), no. 3, 135–162. 10.32523/2306-6172-2024-12-3-135-162Suche in Google Scholar
[16] A. Sá Barreto, Interactions of semilinear progressing waves in two or more space dimensions, Inverse Probl. Imaging 14 (2020), no. 6, 1057–1105. 10.3934/ipi.2020055Suche in Google Scholar
[17] A. Sá Barreto and P. Stefanov, Recovery of a cubic non-linearity in the wave equation in the weakly non-linear regime, Comm. Math. Phys. 392 (2022), no. 1, 25–53. 10.1007/s00220-022-04359-0Suche in Google Scholar
[18] G. Uhlmann and J. Zhai, On an inverse boundary value problem for a nonlinear elastic wave equation, J. Math. Pures Appl. (9) 153 (2021), 114–136. 10.1016/j.matpur.2021.07.005Suche in Google Scholar
[19] Y. Wang and T. Zhou, Inverse problems for quadratic derivative nonlinear wave equations, Comm. Partial Differential Equations 44 (2019), no. 11, 1140–1158. 10.1080/03605302.2019.1612908Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- An inverse problem of finding a time-dependent parameter in a bilinear heat equation
- A novel method for computing core-EP inverse through elementary transformation
- Convergence rates for Tikhonov regularization of a coefficient identification problem
- Carleman estimate for stochastic degenerate wave equation with drift and its application
- Inverse initial problem under Nash strategy for stochastic reaction-diffusion equations with dynamic boundary conditions
- Understanding edge artifacts of the OSEM algorithm in emission tomography
- Unique reconstruction of the inverse spectral problem with mixed data for AKNS operator
- An inverse problem for nonlinear electrodynamic equations
Artikel in diesem Heft
- Frontmatter
- An inverse problem of finding a time-dependent parameter in a bilinear heat equation
- A novel method for computing core-EP inverse through elementary transformation
- Convergence rates for Tikhonov regularization of a coefficient identification problem
- Carleman estimate for stochastic degenerate wave equation with drift and its application
- Inverse initial problem under Nash strategy for stochastic reaction-diffusion equations with dynamic boundary conditions
- Understanding edge artifacts of the OSEM algorithm in emission tomography
- Unique reconstruction of the inverse spectral problem with mixed data for AKNS operator
- An inverse problem for nonlinear electrodynamic equations