Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D

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February 2013, 7(1): 183-197. doi: 10.3934/ipi.2013.7.183

Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D

Georgios Fotopoulos ^1, , Markus Harju ^1, and Valery Serov ^1,

1.	Department of Mathematical Sciences, University of Oulu, PO Box 3000, FIN-90014 Oulu, Finland, Finland, Finland

Received June 2012 Revised November 2012 Published February 2013

Abstract
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We investigate two inverse scattering problems for the nonlinear Schrödinger equation $$ -\Delta u(x) + h(x,|u(x)|)u(x) = k^{2}u(x), \quad x \in \mathbb{R}^2, $$ where $h$ is a very general and possibly singular combination of potentials. The method of Born approximation is applied for the recovery of local singularities and jumps from fixed angle scattering and backscattering data.

Keywords: Schrödinger operator, nonlinearity, Born approximation., inverse problem, backscattering, fixed angle.

Mathematics Subject Classification: 35P25, 35R3.

Citation: Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183

References:

[1]	J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction,", Grundlehren der Mathematischen Wissenschaften, (1976). Google Scholar
[2]	G. Eskin and J. Ralston, Inverse backscattering in two dimensions,, Comm. Math. Phys., 138 (1991), 451. Google Scholar
[3]	L. Grafakos, "Classical and Modern Fourier Analysis,", Pearson Education, (2004). Google Scholar
[4]	R. P. Kanwal, "Generalized Functions. Theory and Applications,", $3^{rd}$ edition, (2004). doi: 10.1007/978-0-8176-8174-6. Google Scholar
[5]	A. Lechleiter, Explicit characterization of the support of non-linear inclusions,, Inverse Probl. Imaging, 5 (2011), 675. doi: 10.3934/ipi.2011.5.675. Google Scholar
[6]	K. Leung, Scattering of transverse-electric electromagnetic waves with a finite nonlinear film,, J. Opt. Soc. Am. B, 5 (1988), 571. Google Scholar
[7]	K. Leung, Exact results for the scattering of electromagnetic waves with a nonlinear film,, Phys. Rev. B, 39 (1989), 3590. Google Scholar
[8]	P. Ola, L. Päivärinta and V. Serov, Recovering singularities from backscattering in two dimensions,, Comm. Partial Differential Equations, 26 (2001), 697. doi: 10.1081/PDE-100001768. Google Scholar
[9]	L. Päivärinta and V. Serov, Recovery of singularities of a multidimensional scattering potential,, SIAM J. Math. Anal., 29 (1998), 697. doi: 10.1137/S0036141096305796. Google Scholar
[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. doi: 10.1088/0266-5611/17/5/306. Google Scholar
[11]	L. Päivärinta and V. Serov, An n-dimensional Borg-Levinson theorem for singular potentials,, Adv. Appl. Math., 29 (2002), 509. doi: 10.1016/S0196-8858(02)00027-1. Google Scholar
[12]	R. T. Prosser, Formal solutions of inverse scattering problems. IV. Error estimates,, J. Math. Phys., 23 (1982), 2127. doi: 10.1063/1.525267. Google Scholar
[13]	J. M. Reyes, Inverse backscattering for the Schrödinger equation in 2D,, Inverse Problems, 23 (2007), 625. doi: 10.1088/0266-5611/23/2/010. Google Scholar
[14]	A. Ruiz, Recovery of the singularities of a potential from fixed angle scattering data,, Comm. Partial Differential Equations, 26 (2001), 1721. doi: 10.1081/PDE-100107457. Google Scholar
[15]	A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data,, Comm. Partial Differential Equations, 30 (2005), 67. doi: 10.1081/PDE-200044450. Google Scholar
[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. Google Scholar
[17]	H. Schürmann and R. Schmoldt, Optical response of a nonlinear absorbing dielectric film,, Opt. Lett., 21 (1996), 387. Google Scholar
[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. Google Scholar
[19]	V. Serov, Inverse fixed angle scattering and backscattering problems in two dimensions,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/6/065002. Google Scholar
[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). doi: 10.1088/1751-8113/43/32/325206. Google Scholar
[21]	V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D,, J. Math. Phys. 53 (2012) 123522., 53 (2012). doi: 10.1063/1.4769825. Google Scholar
[22]	P. Stefanov, Generic uniqueness for two inverse problems in potential scattering,, Comm. Partial Differential Equations, 17 (1992), 55. doi: 10.1080/03605309208820834. Google Scholar

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References:

[1]	J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction,", Grundlehren der Mathematischen Wissenschaften, (1976). Google Scholar
[2]	G. Eskin and J. Ralston, Inverse backscattering in two dimensions,, Comm. Math. Phys., 138 (1991), 451. Google Scholar
[3]	L. Grafakos, "Classical and Modern Fourier Analysis,", Pearson Education, (2004). Google Scholar
[4]	R. P. Kanwal, "Generalized Functions. Theory and Applications,", $3^{rd}$ edition, (2004). doi: 10.1007/978-0-8176-8174-6. Google Scholar
[5]	A. Lechleiter, Explicit characterization of the support of non-linear inclusions,, Inverse Probl. Imaging, 5 (2011), 675. doi: 10.3934/ipi.2011.5.675. Google Scholar
[6]	K. Leung, Scattering of transverse-electric electromagnetic waves with a finite nonlinear film,, J. Opt. Soc. Am. B, 5 (1988), 571. Google Scholar
[7]	K. Leung, Exact results for the scattering of electromagnetic waves with a nonlinear film,, Phys. Rev. B, 39 (1989), 3590. Google Scholar
[8]	P. Ola, L. Päivärinta and V. Serov, Recovering singularities from backscattering in two dimensions,, Comm. Partial Differential Equations, 26 (2001), 697. doi: 10.1081/PDE-100001768. Google Scholar
[9]	L. Päivärinta and V. Serov, Recovery of singularities of a multidimensional scattering potential,, SIAM J. Math. Anal., 29 (1998), 697. doi: 10.1137/S0036141096305796. Google Scholar
[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. doi: 10.1088/0266-5611/17/5/306. Google Scholar
[11]	L. Päivärinta and V. Serov, An n-dimensional Borg-Levinson theorem for singular potentials,, Adv. Appl. Math., 29 (2002), 509. doi: 10.1016/S0196-8858(02)00027-1. Google Scholar
[12]	R. T. Prosser, Formal solutions of inverse scattering problems. IV. Error estimates,, J. Math. Phys., 23 (1982), 2127. doi: 10.1063/1.525267. Google Scholar
[13]	J. M. Reyes, Inverse backscattering for the Schrödinger equation in 2D,, Inverse Problems, 23 (2007), 625. doi: 10.1088/0266-5611/23/2/010. Google Scholar
[14]	A. Ruiz, Recovery of the singularities of a potential from fixed angle scattering data,, Comm. Partial Differential Equations, 26 (2001), 1721. doi: 10.1081/PDE-100107457. Google Scholar
[15]	A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data,, Comm. Partial Differential Equations, 30 (2005), 67. doi: 10.1081/PDE-200044450. Google Scholar
[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. Google Scholar
[17]	H. Schürmann and R. Schmoldt, Optical response of a nonlinear absorbing dielectric film,, Opt. Lett., 21 (1996), 387. Google Scholar
[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. Google Scholar
[19]	V. Serov, Inverse fixed angle scattering and backscattering problems in two dimensions,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/6/065002. Google Scholar
[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). doi: 10.1088/1751-8113/43/32/325206. Google Scholar
[21]	V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D,, J. Math. Phys. 53 (2012) 123522., 53 (2012). doi: 10.1063/1.4769825. Google Scholar
[22]	P. Stefanov, Generic uniqueness for two inverse problems in potential scattering,, Comm. Partial Differential Equations, 17 (1992), 55. doi: 10.1080/03605309208820834. Google Scholar

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