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

Previous Article
Constrained SART algorithm for inverse problems in image reconstruction
IPI Home
This Issue
Next Article
Inverse problem for a coupled parabolic system with discontinuous conductivities: One-dimensional case

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
Related Papers

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 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.

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.

[1]	Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems and Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169
[2]	Ru-Yu Lai. Global uniqueness for an inverse problem for the magnetic Schrödinger operator. Inverse Problems and Imaging, 2011, 5 (1) : 59-73. doi: 10.3934/ipi.2011.5.59
[3]	Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems and Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034
[4]	Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
[5]	Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102
[6]	Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933
[7]	Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems and Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959
[8]	Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems and Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469
[9]	Kaitlyn (Voccola) Muller. A reproducing kernel Hilbert space framework for inverse scattering problems within the Born approximation. Inverse Problems and Imaging, 2019, 13 (6) : 1327-1348. doi: 10.3934/ipi.2019058
[10]	Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123
[11]	Nakao Hayashi, Tohru Ozawa. Schrödinger equations with nonlinearity of integral type. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 475-484. doi: 10.3934/dcds.1995.1.475
[12]	Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations and Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016
[13]	Nguyen Dinh Cong, Roberta Fabbri. On the spectrum of the one-dimensional Schrödinger operator. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541
[14]	Shuyang Dai, Fengru Wang, Jerry Zhijian Yang, Cheng Yuan. On the Cauchy-Born approximation at finite temperature for alloys. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3131-3153. doi: 10.3934/dcdsb.2021176
[15]	Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
[16]	Haidong Liu, Leiga Zhao. Existence results for quasilinear Schrödinger equations with a general nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3429-3444. doi: 10.3934/cpaa.2020059
[17]	Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613
[18]	François Genoud, Charles A. Stuart. Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 137-186. doi: 10.3934/dcds.2008.21.137
[19]	Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266
[20]	Soohyun Bae, Jaeyoung Byeon. Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity. Communications on Pure and Applied Analysis, 2013, 12 (2) : 831-850. doi: 10.3934/cpaa.2013.12.831

2021 Impact Factor: 1.483

Tools

Metrics

Other articles
by authors

on AIMS
Georgios Fotopoulos Markus Harju Valery Serov
on Google Scholar
Georgios Fotopoulos Markus Harju Valery Serov

[Back to Top]