Skip to main content
SpringerLink
Log in

Inverse Problem for the Wave Equation with a Polynomial Nonlinearity

  • Published:
Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

For the wave equation containing a nonlinearity in the form of an \( n \)th order polynomial, we study the problem of determining the coefficients of the polynomial depending on the variable \( x\in \mathbb {R}^3 \). We consider plane waves that propagate in a homogeneous medium in the direction of a unit vector \( \boldsymbol \nu \) with a sharp front and incident on an inhomogeneity localized inside a certain ball \( B(R) \). It is assumed that the solutions of the problems can be measured at the points of the boundary of this ball at the instants of time close to the arrival of the wavefront for all possible values of the vector \( \boldsymbol \nu \). It is shown that the solution of the inverse problem is reduced to a series of X-ray tomography problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. Y. Kurylev, M. Lassas, and G. Uhlmann, “Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations,” Invent. Math. 212, 781–857 (2018). [math.DG], September 20, 2017.

  2. M. Lassas, G. Uhlmann, and Y. Wang, “Inverse problems for semilinear wave equations on Lorentzian manifolds,” Commun. Math. Phys. 360, 555–609 (2018). [math.AP] June 20, 2016.

  3. M. Lassas, “Inverse problems for linear and non-linear hyperbolic equations,” Proc. Int. Congr. Math. 3, 3739–3760 (2018).

    MATH  Google Scholar 

  4. Y. Wang and T. Zhou, “Inverse problems for quadratic derivative nonlinear wave equations,” Commun. Partial Differ. Equ. 44 (11), 1140–1158 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Hintz and G. Uhlmann, “Reconstruction of Lorentzian manifolds from boundary light observation sets,” Int. Math. Res. Notices 22, 6949–6987 (2019). [math.DG] May 27, 2020.

  6. A. S. Barreto, “Interactions of semilinear progressing waves in two or more space dimensions,” Inverse Probl. Imaging 14 (6), 1057–1105 (2020). [math.AP] January 29, 2020.

  7. G. Uhlmann and J. Zhai, “On an inverse boundary value problem for a nonlinear elastic wave equation,” J. Math. Pures Appl. 153, 114–136 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. S. Barreto and P. Stefanov, “Recovery of a general nonlinearity in the semilinear wave equation,” [math.AP] July 18, 2021.

  9. P. Hintz, G. Uhlmann, and J. Zhai, “The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds” (2021). [math.AP] March 15, 2021.

  10. A. S. Barreto and P. Stefanov, “Recovery of a cubic non-linearity in the wave equation in the weakly non-linear regime,” Commun. Math. Phys. 392, 25–53 (2022). https://doi.org/10.1007/s00220-022-04359-0

  11. V. G. Romanov and T. V. Bugueva, “Inverse problem for the nonlinear wave equation,” Sib. Zh. Ind. Mat. 25 (2), 83–100 (2022) [in Russian].

    MathSciNet  Google Scholar 

  12. V. G. Romanov and T. V. Bugueva, “The problem of determining the coefficient multiplying the nonlinear term in a quasilinear wave equation,” Sib. Zh. Ind. Mat. 25 (3), 154–169 (2022) [in Russian].

    Google Scholar 

  13. V. G. Romanov, “An inverse problem for a semilinear wave equation,” Dokl. Ross. Akad. Nauk. Mat. Inf. Protsessy Upr. 504 (1), 36–41 (2022) [Dokl. Math. 105 (3), 166–170 (2022)].

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986; Mir, Moscow, 1990).

    Book  MATH  Google Scholar 

  15. M. E. Davison, “A singular value decomposition for the Radon transform in \( n \)-dimensional Euclidean space,” Numer. Funct. Anal. Optim. 3, 321–340 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. V. Pikalov and N. G. Preobrazhenskii, “Computer-aided tomography and physical experiment,” Usp. Fiz. Nauk 141 (3), 469–498 (1983) [Phys.–Usp. 26 (11), 974–990 (1983)].

    Google Scholar 

  17. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

    MATH  Google Scholar 

  18. A. K. Louis, “Orthogonal function series expansions and the null space of the Radon transform,” SIAM J. Math. Anal. 15 (3), 621–633 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. K. Louis, “Incomplete data problems in x-ray computerized tomography,” Numer. Math. 48 (3), 251–262, (1986). https://doi.org/10.1007/BF01389474

  20. A. N. Tikhonov, V. Ya. Arsenin, and A. A. Timonov, Mathematical Problems of Computer Tomography (Nauka, Moscow, 1987) [in Russian].

    MATH  Google Scholar 

  21. A. G. Michette and C. J. Buckley, X-Ray Science and Technology (Taylor & Francis, New York, 1993).

    Google Scholar 

  22. V. M. Boiko, A. M. Orishich, A. A. Pavlov, and V. V. Pikalov, Methods of Optical Diagnostics in Aerophysical Experiment (Izd. NGU, Novosibirsk, 2009) [in Russian].

    Google Scholar 

  23. E. Yu. Derevtsov, A. V. Efimov, A. K. Louis, and T. Schuster, “Singular value decomposition and its application to numerical inversion for ray transforms in 2D vector tomography,” J. Inverse Ill-Posed Probl. 19 (4–5), 689–715 (2011). https://doi.org/10.1515/jiip.2011.047

  24. D. S. Anikonov, V. G. Nazarov, and I. V. Prokhorov, “An integrodifferential indicator for the problem of single beam tomography,” Sib. Zh. Ind. Mat. 17 (2), 3–10 (2014) [J. Appl. Ind. Math. 8 (3), 301–306 (2014)].

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Liu, H. Yu, and H. Yu, “Singular value decomposition-based 2D image reconstruction for computed tomography,” J. X-Ray Sci. Technol. 25 (1), 1–22 (2016). https://doi.org/10.3233/XST-16173

  26. L. Borg, J. Frikel, J. S. Orgensen, and E. T. Quinto, “Analyzing reconstruction artifacts from arbitrary incomplete X-ray CT data.” [math.FA] June 26, 2018.

  27. Y. Xu, H. Yu, A. Sushmit, Q. Lyu, G. Wang, Y. Li, X. Cao, and J. S. Maltz, “Cardiac CT motion artifact grading via semi-automatic labeling and vessel tracking using synthetic image-augmented training data,” J. X-Ray Sci. Technol. 30 (3), 433–445 (2022). https://doi.org/10.3233/XST-211109

  28. V. A. Sharafutdinov, “On determining an optical body located in a homogeneous medium based on its images,” in Mathematical Methods for Solving Direct and Inverse Problems of Geophysics (Izd. VTs SO AN SSSR, Novosibirsk, 1981), 123–148 [in Russian].

Download references

Funding

The work was carried out within the framework of the state assignment for Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, project no. FWNF-2022-0009.

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    V. G. Romanov & T. V. Bugueva

  2. Novosibirsk State University, Novosibirsk, 630090, Russia

    T. V. Bugueva

Authors
  1. V. G. Romanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. V. Bugueva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to V. G. Romanov or T. V. Bugueva.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Romanov, V.G., Bugueva, T.V. Inverse Problem for the Wave Equation with a Polynomial Nonlinearity. J. Appl. Ind. Math. 17, 163–167 (2023). https://doi.org/10.1134/S1990478923010180

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478923010180

Keywords

Search

Navigation