January  2021, 41(1): 455-469. doi: 10.3934/dcds.2020380

Inverse problems for nonlinear hyperbolic equations

1. 

Department of Mathematics, University of Washington, Seattle, WA 98195, USA, Institute for Advanced Study, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China

2. 

Institute for Advanced Study, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China

* Corresponding author: gunther@math.washington.edu

Received  April 2020 Published  January 2021 Early access  November 2020

Fund Project: The first author was partially supported by NSF, a Walker Professorship at UW and a Si-Yuan Professorship at IAS, HKUST

There has been considerable progress in recent years in solving inverse problems for nonlinear hyperbolic equations. One of the striking aspects of these developments is the use of nonlinearity to get new information, which is not possible for the corresponding linear equations. We illustrate this for several examples including Einstein equations and the equations of nonlinear elasticity among others.

Citation: Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380
References:
[1]

M. AndersonA. KatsudaY. KurylevM. Lassas and M. Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem, Invent. Math., 158 (2004), 261-321.  doi: 10.1007/s00222-004-0371-6.  Google Scholar

[2]

M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.   Google Scholar

[3]

A. N. Bernal and M. Sánchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Comm. Math. Phys., 257 (2005), 43-50.  doi: 10.1007/s00220-005-1346-1.  Google Scholar

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S. Bhattacharyya, Local uniqueness of the density from partial boundary data for isotropic elastodynamics, Inverse Problems, 34 (2018), 125001, 10 pp. doi: 10.1088/1361-6420/aade76.  Google Scholar

[5]

X. Chen, M. Lassas, L. Oksanen and G. P. Paternain, Detection of Hermitian connections in wave equations with cubic non-linearity, preprint, arXiv: 1902.05711, 2019. Google Scholar

[6]

M. de HoopG. Uhlmann and Y. Wang, Nonlinear responses from the interaction of two progressing waves at an interface, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 347-363.  doi: 10.1016/j.anihpc.2018.04.005.  Google Scholar

[7]

M. V. de Hoop, G. Uhlmann and Y. Wang, Nonlinear interaction of waves in elastodynamics and an inverse problem, Mathematische Annalen, 376 (2020), 765-795. doi: 10.1007/s00208-018-01796-y.  Google Scholar

[8]

W. De Lima and M. Hamilton, Finite-amplitude waves in isotropic elastic plates, Journal of Sound and Vibration, 265 (2003), 819-839.   Google Scholar

[9]

A. Feizmohammadi, J. Ilmavirta, Y. Kian and L. Oksanen, Recovery of time dependent coefficients from boundary data for hyperbolic equations, preprint, arXiv: 1901.04211, 2019. Google Scholar

[10]

A. Feizmohammadi and L. Oksanen, Recovery of zeroth order coefficients in non-linear wave equations, preprint, arXiv: 1903.12636, 2019. Google Scholar

[11]

A. Feizmohammadi and L. Oksanen, An inverse problem for a semi-linear elliptic equation in Riemannian geometries, J. Differential Equations, 269 (2020), 4683-4719.  doi: 10.1016/j.jde.2020.03.037.  Google Scholar

[12]

V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267.  doi: 10.1215/S0012-7094-81-04814-6.  Google Scholar

[13]

S. Hansen and G. Uhlmann, Propagation of polarization in elastodynamics with residual stress and travel times, Math. Ann., 326 (2003), 563-587.  doi: 10.1007/s00208-003-0437-6.  Google Scholar

[14]

P. Hintz and G. Uhlmann, Reconstruction of Lorentzian manifolds from boundary light observation sets, Int. Math. Res. Not. IMRN, 2019 (2019), 6949-6987.  doi: 10.1093/imrn/rnx320.  Google Scholar

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V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12.  doi: 10.1007/BF00392201.  Google Scholar

[16] A. KachalovY. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, CRC Press, 2001.  doi: 10.1201/9781420036220.  Google Scholar
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Y. Kurylev, M. Lassas, L. Oksanen and G. Uhlmann, Inverse problem for Einstein-scalar field equations, preprint, arXiv: 1406.4776, 2014. Google Scholar

[18]

Y. KurylevM. Lassas and G. Uhlmann, Rigidity of broken geodesic flow and inverse problems, Amer. J. Math., 132 (2010), 529-562.  doi: 10.1353/ajm.0.0103.  Google Scholar

[19]

Y. Kurylev, M. Lassas and G. Uhlmann, Inverse problems in spacetime I: Inverse problems for Einstein equations-extended preprint version, preprint, arXiv: 1405.4503, 2014. Google Scholar

[20]

Y. KurylevM. Lassas and G. Uhlmann, Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Invent. Math., 212 (2018), 781-857.  doi: 10.1007/s00222-017-0780-y.  Google Scholar

[21]

L. Landau, E. Lifshitz, J. Sykes, W. Reid and E. H. Dill, Theory of elasticity: Vol. 7 of course of theoretical physics, Phys. Today, 13 (1960), 44. Google Scholar

[22]

M. Lassas, G. Uhlmann and Y. Wang, Determination of vacuum space-times from the Einstein-Maxwell equations, preprint, arXiv: 1703.10704, 2017. Google Scholar

[23]

M. LassasG. Uhlmann and Y. Wang, Inverse problems for semilinear wave equations on Lorentzian manifolds, Comm. Math. Phys., 360 (2018), 555-609.  doi: 10.1007/s00220-018-3135-7.  Google Scholar

[24]

X. Li and J.-N. Wang, Determination of viscosity in the stationary Navier-Stokes equations, J. Differential Equations, 242 (2007), 24-39.  doi: 10.1016/j.jde.2007.07.008.  Google Scholar

[25]

R. B. Melrose and G. A. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.  doi: 10.1002/cpa.3160320403.  Google Scholar

[26]

G. P. PaternainM. SaloG. Uhlmann and H. Zhou, The geodesic X-ray transform with matrix weights, Amer. J. Math., 141 (2019), 1707-1750.  doi: 10.1353/ajm.2019.0045.  Google Scholar

[27]

L. Rachele, An inverse problem in elastodynamics: Uniqueness of the wave speeds in the interior, J. Differential Equations, 162 (2000), 300-325.  doi: 10.1006/jdeq.1999.3657.  Google Scholar

[28]

L. V. Rachele, Uniqueness of the density in an inverse problem for isotropic elastodynamics, Trans. Amer. Math. Soc., 355 (2003), 4781-4806.  doi: 10.1090/S0002-9947-03-03268-9.  Google Scholar

[29]

P. Stefanov and G. Uhlmann, Stable determination of the hyperbolic Dirichlet-to-Neumann map for generic simple metrics, Int. Math. Res. Not., 17 (2005), 1047-1061.  doi: 10.1155/IMRN.2005.1047.  Google Scholar

[30]

P. Stefanov, G. Uhlmann and A. Vasy, Local recovery of the compressional and shear speeds from the hyperbolic DN map, Inverse Problems, 34 (2018), 014003, 13 pp. doi: 10.1088/1361-6420/aa9833.  Google Scholar

[31]

P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, Anal. PDE, 11 (2018), 1381-1414.  doi: 10.2140/apde.2018.11.1381.  Google Scholar

[32]

Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119 (1997), 771-797.  doi: 10.1353/ajm.1997.0027.  Google Scholar

[33]

D. Tataru, Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884.  doi: 10.1080/03605309508821117.  Google Scholar

[34]

K. K. Uhlenbeck, Variational problems for gauge fields, Proceedings of the International Congress of Mathematicians, (1984), 585-591.  Google Scholar

[35]

G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform, Invent. Math., 205 (2016), 83-120.  doi: 10.1007/s00222-015-0631-7.  Google Scholar

[36]

G. Uhlmann and Y. Wang, Convolutional neural networks in phase space and inverse problems, preprint, arXiv: 1811.04022, 2018. Google Scholar

[37]

G. Uhlmann and Y. Wang, Determination of space-time structures from gravitational perturbations, Comm. Pure Appl. Math., 73 (2020), 1315-1367.  doi: 10.1002/cpa.21882.  Google Scholar

[38]

G. Uhlmann and J. Zhai, On an inverse boundary value problem for a nonlinear elastic wave equation, preprint, arXiv: 1912.11756, 2019. Google Scholar

[39]

Y. Wang and T. Zhou, Inverse problems for quadratic derivative nonlinear wave equations, Comm. Partial Differential Equations, 44 (2019), 1140-1158.  doi: 10.1080/03605302.2019.1612908.  Google Scholar

[40]

G. B. Whitham, Linear and Nonlinear Waves, volume 42, John Wiley & Sons, 2011. Google Scholar

show all references

References:
[1]

M. AndersonA. KatsudaY. KurylevM. Lassas and M. Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem, Invent. Math., 158 (2004), 261-321.  doi: 10.1007/s00222-004-0371-6.  Google Scholar

[2]

M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.   Google Scholar

[3]

A. N. Bernal and M. Sánchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Comm. Math. Phys., 257 (2005), 43-50.  doi: 10.1007/s00220-005-1346-1.  Google Scholar

[4]

S. Bhattacharyya, Local uniqueness of the density from partial boundary data for isotropic elastodynamics, Inverse Problems, 34 (2018), 125001, 10 pp. doi: 10.1088/1361-6420/aade76.  Google Scholar

[5]

X. Chen, M. Lassas, L. Oksanen and G. P. Paternain, Detection of Hermitian connections in wave equations with cubic non-linearity, preprint, arXiv: 1902.05711, 2019. Google Scholar

[6]

M. de HoopG. Uhlmann and Y. Wang, Nonlinear responses from the interaction of two progressing waves at an interface, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 347-363.  doi: 10.1016/j.anihpc.2018.04.005.  Google Scholar

[7]

M. V. de Hoop, G. Uhlmann and Y. Wang, Nonlinear interaction of waves in elastodynamics and an inverse problem, Mathematische Annalen, 376 (2020), 765-795. doi: 10.1007/s00208-018-01796-y.  Google Scholar

[8]

W. De Lima and M. Hamilton, Finite-amplitude waves in isotropic elastic plates, Journal of Sound and Vibration, 265 (2003), 819-839.   Google Scholar

[9]

A. Feizmohammadi, J. Ilmavirta, Y. Kian and L. Oksanen, Recovery of time dependent coefficients from boundary data for hyperbolic equations, preprint, arXiv: 1901.04211, 2019. Google Scholar

[10]

A. Feizmohammadi and L. Oksanen, Recovery of zeroth order coefficients in non-linear wave equations, preprint, arXiv: 1903.12636, 2019. Google Scholar

[11]

A. Feizmohammadi and L. Oksanen, An inverse problem for a semi-linear elliptic equation in Riemannian geometries, J. Differential Equations, 269 (2020), 4683-4719.  doi: 10.1016/j.jde.2020.03.037.  Google Scholar

[12]

V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267.  doi: 10.1215/S0012-7094-81-04814-6.  Google Scholar

[13]

S. Hansen and G. Uhlmann, Propagation of polarization in elastodynamics with residual stress and travel times, Math. Ann., 326 (2003), 563-587.  doi: 10.1007/s00208-003-0437-6.  Google Scholar

[14]

P. Hintz and G. Uhlmann, Reconstruction of Lorentzian manifolds from boundary light observation sets, Int. Math. Res. Not. IMRN, 2019 (2019), 6949-6987.  doi: 10.1093/imrn/rnx320.  Google Scholar

[15]

V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12.  doi: 10.1007/BF00392201.  Google Scholar

[16] A. KachalovY. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, CRC Press, 2001.  doi: 10.1201/9781420036220.  Google Scholar
[17]

Y. Kurylev, M. Lassas, L. Oksanen and G. Uhlmann, Inverse problem for Einstein-scalar field equations, preprint, arXiv: 1406.4776, 2014. Google Scholar

[18]

Y. KurylevM. Lassas and G. Uhlmann, Rigidity of broken geodesic flow and inverse problems, Amer. J. Math., 132 (2010), 529-562.  doi: 10.1353/ajm.0.0103.  Google Scholar

[19]

Y. Kurylev, M. Lassas and G. Uhlmann, Inverse problems in spacetime I: Inverse problems for Einstein equations-extended preprint version, preprint, arXiv: 1405.4503, 2014. Google Scholar

[20]

Y. KurylevM. Lassas and G. Uhlmann, Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Invent. Math., 212 (2018), 781-857.  doi: 10.1007/s00222-017-0780-y.  Google Scholar

[21]

L. Landau, E. Lifshitz, J. Sykes, W. Reid and E. H. Dill, Theory of elasticity: Vol. 7 of course of theoretical physics, Phys. Today, 13 (1960), 44. Google Scholar

[22]

M. Lassas, G. Uhlmann and Y. Wang, Determination of vacuum space-times from the Einstein-Maxwell equations, preprint, arXiv: 1703.10704, 2017. Google Scholar

[23]

M. LassasG. Uhlmann and Y. Wang, Inverse problems for semilinear wave equations on Lorentzian manifolds, Comm. Math. Phys., 360 (2018), 555-609.  doi: 10.1007/s00220-018-3135-7.  Google Scholar

[24]

X. Li and J.-N. Wang, Determination of viscosity in the stationary Navier-Stokes equations, J. Differential Equations, 242 (2007), 24-39.  doi: 10.1016/j.jde.2007.07.008.  Google Scholar

[25]

R. B. Melrose and G. A. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.  doi: 10.1002/cpa.3160320403.  Google Scholar

[26]

G. P. PaternainM. SaloG. Uhlmann and H. Zhou, The geodesic X-ray transform with matrix weights, Amer. J. Math., 141 (2019), 1707-1750.  doi: 10.1353/ajm.2019.0045.  Google Scholar

[27]

L. Rachele, An inverse problem in elastodynamics: Uniqueness of the wave speeds in the interior, J. Differential Equations, 162 (2000), 300-325.  doi: 10.1006/jdeq.1999.3657.  Google Scholar

[28]

L. V. Rachele, Uniqueness of the density in an inverse problem for isotropic elastodynamics, Trans. Amer. Math. Soc., 355 (2003), 4781-4806.  doi: 10.1090/S0002-9947-03-03268-9.  Google Scholar

[29]

P. Stefanov and G. Uhlmann, Stable determination of the hyperbolic Dirichlet-to-Neumann map for generic simple metrics, Int. Math. Res. Not., 17 (2005), 1047-1061.  doi: 10.1155/IMRN.2005.1047.  Google Scholar

[30]

P. Stefanov, G. Uhlmann and A. Vasy, Local recovery of the compressional and shear speeds from the hyperbolic DN map, Inverse Problems, 34 (2018), 014003, 13 pp. doi: 10.1088/1361-6420/aa9833.  Google Scholar

[31]

P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, Anal. PDE, 11 (2018), 1381-1414.  doi: 10.2140/apde.2018.11.1381.  Google Scholar

[32]

Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119 (1997), 771-797.  doi: 10.1353/ajm.1997.0027.  Google Scholar

[33]

D. Tataru, Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884.  doi: 10.1080/03605309508821117.  Google Scholar

[34]

K. K. Uhlenbeck, Variational problems for gauge fields, Proceedings of the International Congress of Mathematicians, (1984), 585-591.  Google Scholar

[35]

G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform, Invent. Math., 205 (2016), 83-120.  doi: 10.1007/s00222-015-0631-7.  Google Scholar

[36]

G. Uhlmann and Y. Wang, Convolutional neural networks in phase space and inverse problems, preprint, arXiv: 1811.04022, 2018. Google Scholar

[37]

G. Uhlmann and Y. Wang, Determination of space-time structures from gravitational perturbations, Comm. Pure Appl. Math., 73 (2020), 1315-1367.  doi: 10.1002/cpa.21882.  Google Scholar

[38]

G. Uhlmann and J. Zhai, On an inverse boundary value problem for a nonlinear elastic wave equation, preprint, arXiv: 1912.11756, 2019. Google Scholar

[39]

Y. Wang and T. Zhou, Inverse problems for quadratic derivative nonlinear wave equations, Comm. Partial Differential Equations, 44 (2019), 1140-1158.  doi: 10.1080/03605302.2019.1612908.  Google Scholar

[40]

G. B. Whitham, Linear and Nonlinear Waves, volume 42, John Wiley & Sons, 2011. Google Scholar

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