# American Institute of Mathematical Sciences

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

 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

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.
Citation: Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems and 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, 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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##### 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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