American Institute of Mathematical Sciences

Advanced
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

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

show all references

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

[1]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541
[14]

Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
[15]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613
[16]

François Genoud, Charles A. Stuart. Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 137-186. doi: 10.3934/dcds.2008.21.137
[17]

Soohyun Bae, Jaeyoung Byeon. Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 831-850. doi: 10.3934/cpaa.2013.12.831
[18]

Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266
[19]

Haidong Liu, Leiga Zhao. Existence results for quasilinear Schrödinger equations with a general nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3429-3444. doi: 10.3934/cpaa.2020059
[20]

Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences