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 & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183
References:
[1]

Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[2]

Comm. Math. Phys., 138 (1991), 451-486.  Google Scholar

[3]

Pearson Education, Inc., Upper Saddle River, New Jersey, 2004.  Google Scholar

[4]

$3^{rd}$ edition, Birkhäuser Boston, Inc., Boston, 2004. doi: 10.1007/978-0-8176-8174-6.  Google Scholar

[5]

Inverse Probl. Imaging, 5 (2011), 675-694. doi: 10.3934/ipi.2011.5.675.  Google Scholar

[6]

J. Opt. Soc. Am. B, 5 (1988), 571-574. Google Scholar

[7]

Phys. Rev. B, 39 (1989), 3590-3598. Google Scholar

[8]

Comm. Partial Differential Equations, 26 (2001), 697-715. doi: 10.1081/PDE-100001768.  Google Scholar

[9]

SIAM J. Math. Anal., 29 (1998), 697-711. doi: 10.1137/S0036141096305796.  Google Scholar

[10]

Inverse Problems, 17 (2001), 1321-1326. doi: 10.1088/0266-5611/17/5/306.  Google Scholar

[11]

Adv. Appl. Math., 29 (2002), 509-520. doi: 10.1016/S0196-8858(02)00027-1.  Google Scholar

[12]

J. Math. Phys., 23 (1982), 2127-2130. doi: 10.1063/1.525267.  Google Scholar

[13]

Inverse Problems, 23 (2007), 625-643. doi: 10.1088/0266-5611/23/2/010.  Google Scholar

[14]

Comm. Partial Differential Equations, 26 (2001), 1721-1738. doi: 10.1081/PDE-100107457.  Google Scholar

[15]

Comm. Partial Differential Equations, 30 (2005), 67-96. doi: 10.1081/PDE-200044450.  Google Scholar

[16]

Z. Phys. B, 92 (1993), 179-186. Google Scholar

[17]

Opt. Lett., 21 (1996), 387-389. Google Scholar

[18]

Dokl. Akad. Nauk, 385 (2002), 160-162.  Google Scholar

[19]

Inverse Problems, 24 (2008), 065002, 14 pp. doi: 10.1088/0266-5611/24/6/065002.  Google Scholar

[20]

J. Phys. A: Math. Theor., 43 (2010), 325206, 15 pp. doi: 10.1088/1751-8113/43/32/325206.  Google Scholar

[21]

J. Math. Phys. 53 (2012) 123522. doi: 10.1063/1.4769825.  Google Scholar

[22]

Comm. Partial Differential Equations, 17 (1992), 55-68. doi: 10.1080/03605309208820834.  Google Scholar

show all references

References:
[1]

Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[2]

Comm. Math. Phys., 138 (1991), 451-486.  Google Scholar

[3]

Pearson Education, Inc., Upper Saddle River, New Jersey, 2004.  Google Scholar

[4]

$3^{rd}$ edition, Birkhäuser Boston, Inc., Boston, 2004. doi: 10.1007/978-0-8176-8174-6.  Google Scholar

[5]

Inverse Probl. Imaging, 5 (2011), 675-694. doi: 10.3934/ipi.2011.5.675.  Google Scholar

[6]

J. Opt. Soc. Am. B, 5 (1988), 571-574. Google Scholar

[7]

Phys. Rev. B, 39 (1989), 3590-3598. Google Scholar

[8]

Comm. Partial Differential Equations, 26 (2001), 697-715. doi: 10.1081/PDE-100001768.  Google Scholar

[9]

SIAM J. Math. Anal., 29 (1998), 697-711. doi: 10.1137/S0036141096305796.  Google Scholar

[10]

Inverse Problems, 17 (2001), 1321-1326. doi: 10.1088/0266-5611/17/5/306.  Google Scholar

[11]

Adv. Appl. Math., 29 (2002), 509-520. doi: 10.1016/S0196-8858(02)00027-1.  Google Scholar

[12]

J. Math. Phys., 23 (1982), 2127-2130. doi: 10.1063/1.525267.  Google Scholar

[13]

Inverse Problems, 23 (2007), 625-643. doi: 10.1088/0266-5611/23/2/010.  Google Scholar

[14]

Comm. Partial Differential Equations, 26 (2001), 1721-1738. doi: 10.1081/PDE-100107457.  Google Scholar

[15]

Comm. Partial Differential Equations, 30 (2005), 67-96. doi: 10.1081/PDE-200044450.  Google Scholar

[16]

Z. Phys. B, 92 (1993), 179-186. Google Scholar

[17]

Opt. Lett., 21 (1996), 387-389. Google Scholar

[18]

Dokl. Akad. Nauk, 385 (2002), 160-162.  Google Scholar

[19]

Inverse Problems, 24 (2008), 065002, 14 pp. doi: 10.1088/0266-5611/24/6/065002.  Google Scholar

[20]

J. Phys. A: Math. Theor., 43 (2010), 325206, 15 pp. doi: 10.1088/1751-8113/43/32/325206.  Google Scholar

[21]

J. Math. Phys. 53 (2012) 123522. doi: 10.1063/1.4769825.  Google Scholar

[22]

Comm. Partial Differential Equations, 17 (1992), 55-68. doi: 10.1080/03605309208820834.  Google Scholar

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