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Inverse problem for a coupled parabolic system with discontinuous conductivities: One-dimensional case
Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D
1. | Department of Mathematical Sciences, University of Oulu, PO Box 3000, FIN-90014 Oulu, Finland, Finland, Finland |
References:
[1] |
J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. |
[2] |
G. Eskin and J. Ralston, Inverse backscattering in two dimensions, Comm. Math. Phys., 138 (1991), 451-486. |
[3] |
L. Grafakos, "Classical and Modern Fourier Analysis," Pearson Education, Inc., Upper Saddle River, New Jersey, 2004. |
[4] |
R. P. Kanwal, "Generalized Functions. Theory and Applications," $3^{rd}$ edition, Birkhäuser Boston, Inc., Boston, 2004.
doi: 10.1007/978-0-8176-8174-6. |
[5] |
A. Lechleiter, Explicit characterization of the support of non-linear inclusions, Inverse Probl. Imaging, 5 (2011), 675-694.
doi: 10.3934/ipi.2011.5.675. |
[6] |
K. Leung, Scattering of transverse-electric electromagnetic waves with a finite nonlinear film, J. Opt. Soc. Am. B, 5 (1988), 571-574. |
[7] |
K. Leung, Exact results for the scattering of electromagnetic waves with a nonlinear film, Phys. Rev. B, 39 (1989), 3590-3598. |
[8] |
P. Ola, L. Päivärinta and V. Serov, Recovering singularities from backscattering in two dimensions, Comm. Partial Differential Equations, 26 (2001), 697-715.
doi: 10.1081/PDE-100001768. |
[9] |
L. Päivärinta and V. Serov, Recovery of singularities of a multidimensional scattering potential, SIAM J. Math. Anal., 29 (1998), 697-711.
doi: 10.1137/S0036141096305796. |
[10] |
L. Päivärinta and V. Serov, New mapping properties for the resolvent of the Laplacian and recovery of singularities of a multi-dimensional scattering potential, Inverse Problems, 17 (2001), 1321-1326.
doi: 10.1088/0266-5611/17/5/306. |
[11] |
L. Päivärinta and V. Serov, An n-dimensional Borg-Levinson theorem for singular potentials, Adv. Appl. Math., 29 (2002), 509-520.
doi: 10.1016/S0196-8858(02)00027-1. |
[12] |
R. T. Prosser, Formal solutions of inverse scattering problems. IV. Error estimates, J. Math. Phys., 23 (1982), 2127-2130.
doi: 10.1063/1.525267. |
[13] |
J. M. Reyes, Inverse backscattering for the Schrödinger equation in 2D, Inverse Problems, 23 (2007), 625-643.
doi: 10.1088/0266-5611/23/2/010. |
[14] |
A. Ruiz, Recovery of the singularities of a potential from fixed angle scattering data, Comm. Partial Differential Equations, 26 (2001), 1721-1738.
doi: 10.1081/PDE-100107457. |
[15] |
A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data, Comm. Partial Differential Equations, 30 (2005), 67-96.
doi: 10.1081/PDE-200044450. |
[16] |
H. Schürmann and R. Schmoldt, On the theory of reflectivity and transmissivity of a lossless nonlinear dielectric slab, Z. Phys. B, 92 (1993), 179-186. |
[17] |
H. Schürmann and R. Schmoldt, Optical response of a nonlinear absorbing dielectric film, Opt. Lett., 21 (1996), 387-389. |
[18] |
V. Serov, Reconstruction of singularities of the potential in the two-dimensional Schrödinger operator from fixed-angle scattering data. (Russian), Dokl. Akad. Nauk, 385 (2002), 160-162. |
[19] |
V. Serov, Inverse fixed angle scattering and backscattering problems in two dimensions, Inverse Problems, 24 (2008), 065002, 14 pp.
doi: 10.1088/0266-5611/24/6/065002. |
[20] |
V. Serov and J. Sandhu, Inverse backscattering problem for the generalized nonlinear Schrödinger operator in two dimensions, J. Phys. A: Math. Theor., 43 (2010), 325206, 15 pp.
doi: 10.1088/1751-8113/43/32/325206. |
[21] |
V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D, J. Math. Phys. 53 (2012) 123522.
doi: 10.1063/1.4769825. |
[22] |
P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations, 17 (1992), 55-68.
doi: 10.1080/03605309208820834. |
show all references
References:
[1] |
J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. |
[2] |
G. Eskin and J. Ralston, Inverse backscattering in two dimensions, Comm. Math. Phys., 138 (1991), 451-486. |
[3] |
L. Grafakos, "Classical and Modern Fourier Analysis," Pearson Education, Inc., Upper Saddle River, New Jersey, 2004. |
[4] |
R. P. Kanwal, "Generalized Functions. Theory and Applications," $3^{rd}$ edition, Birkhäuser Boston, Inc., Boston, 2004.
doi: 10.1007/978-0-8176-8174-6. |
[5] |
A. Lechleiter, Explicit characterization of the support of non-linear inclusions, Inverse Probl. Imaging, 5 (2011), 675-694.
doi: 10.3934/ipi.2011.5.675. |
[6] |
K. Leung, Scattering of transverse-electric electromagnetic waves with a finite nonlinear film, J. Opt. Soc. Am. B, 5 (1988), 571-574. |
[7] |
K. Leung, Exact results for the scattering of electromagnetic waves with a nonlinear film, Phys. Rev. B, 39 (1989), 3590-3598. |
[8] |
P. Ola, L. Päivärinta and V. Serov, Recovering singularities from backscattering in two dimensions, Comm. Partial Differential Equations, 26 (2001), 697-715.
doi: 10.1081/PDE-100001768. |
[9] |
L. Päivärinta and V. Serov, Recovery of singularities of a multidimensional scattering potential, SIAM J. Math. Anal., 29 (1998), 697-711.
doi: 10.1137/S0036141096305796. |
[10] |
L. Päivärinta and V. Serov, New mapping properties for the resolvent of the Laplacian and recovery of singularities of a multi-dimensional scattering potential, Inverse Problems, 17 (2001), 1321-1326.
doi: 10.1088/0266-5611/17/5/306. |
[11] |
L. Päivärinta and V. Serov, An n-dimensional Borg-Levinson theorem for singular potentials, Adv. Appl. Math., 29 (2002), 509-520.
doi: 10.1016/S0196-8858(02)00027-1. |
[12] |
R. T. Prosser, Formal solutions of inverse scattering problems. IV. Error estimates, J. Math. Phys., 23 (1982), 2127-2130.
doi: 10.1063/1.525267. |
[13] |
J. M. Reyes, Inverse backscattering for the Schrödinger equation in 2D, Inverse Problems, 23 (2007), 625-643.
doi: 10.1088/0266-5611/23/2/010. |
[14] |
A. Ruiz, Recovery of the singularities of a potential from fixed angle scattering data, Comm. Partial Differential Equations, 26 (2001), 1721-1738.
doi: 10.1081/PDE-100107457. |
[15] |
A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data, Comm. Partial Differential Equations, 30 (2005), 67-96.
doi: 10.1081/PDE-200044450. |
[16] |
H. Schürmann and R. Schmoldt, On the theory of reflectivity and transmissivity of a lossless nonlinear dielectric slab, Z. Phys. B, 92 (1993), 179-186. |
[17] |
H. Schürmann and R. Schmoldt, Optical response of a nonlinear absorbing dielectric film, Opt. Lett., 21 (1996), 387-389. |
[18] |
V. Serov, Reconstruction of singularities of the potential in the two-dimensional Schrödinger operator from fixed-angle scattering data. (Russian), Dokl. Akad. Nauk, 385 (2002), 160-162. |
[19] |
V. Serov, Inverse fixed angle scattering and backscattering problems in two dimensions, Inverse Problems, 24 (2008), 065002, 14 pp.
doi: 10.1088/0266-5611/24/6/065002. |
[20] |
V. Serov and J. Sandhu, Inverse backscattering problem for the generalized nonlinear Schrödinger operator in two dimensions, J. Phys. A: Math. Theor., 43 (2010), 325206, 15 pp.
doi: 10.1088/1751-8113/43/32/325206. |
[21] |
V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D, J. Math. Phys. 53 (2012) 123522.
doi: 10.1063/1.4769825. |
[22] |
P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations, 17 (1992), 55-68.
doi: 10.1080/03605309208820834. |
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