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 and 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

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

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

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[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.

[36]

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

[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.

[38]

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

[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.

[40]

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

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

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

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

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[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.

[36]

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

[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.

[38]

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

[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.

[40]

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

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