Startseite Mathematik An inverse problem for nonlinear electrodynamic equations
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

An inverse problem for nonlinear electrodynamic equations

  • Vladimir G. Romanov ORCID logo EMAIL logo
Veröffentlicht/Copyright: 17. Juni 2025

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 3 for different ν are used for posing an inverse problem. It is shown that the inverse problem is decomposed in seven separate problems. One of them is the X-ray tomography problem while 6 others are identical one to other integral geometry problems on a family of strait lines with a given weight function. The latter problems are studied and a stability estimate of solutions is found.

MSC 2020: 35F45; 35L40; 35R25

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

Received: 2025-05-19
Accepted: 2025-05-22
Published Online: 2025-06-17
Published in Print: 2025-08-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 10.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jiip-2025-0037/pdf?lang=de
Button zum nach oben scrollen