American Institute of Mathematical Sciences

Advanced
  • Previous Article
    An inverse problem for the magnetic Schrödinger operator on a half space with partial data
  • IPI Home
  • This Issue
  • Next Article
    Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map
November  2014, 8(4): 1151-1167. doi: 10.3934/ipi.2014.8.1151

The nonlinear Fourier transform for two-dimensional subcritical potentials

Michael Music 1,

1. 

Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, United States

Received  November 2013 Revised  June 2014 Published  November 2014

The inverse scattering method for the Novikov-Veselov equation is studied for a larger class of Schrödinger potentials than could be handled previously. Previous work concerns so-called conductivity type potentials, which have a bounded positive solution at zero energy and are a nowhere dense set of potentials. We relax the conductivity type assumption to include logarithmically growing positive solutions at zero energy. These potentials are stable under perturbations. Assuming only that the potential is subcritical and has two weak derivatives in a weighted Sobolev space, we prove that the associated scattering transform can be inverted, and the original potential is recovered from the scattering data.
Keywords: inverse scattering, Schrodinger equation, complex geometric optics., Novikov-Veselov equation.
Mathematics Subject Classification: Primary: 37K15; Secondary: 35J1.
Citation: Michael Music. The nonlinear Fourier transform for two-dimensional subcritical potentials. Inverse Problems & Imaging, 2014, 8 (4) : 1151-1167. doi: 10.3934/ipi.2014.8.1151
References:
[1]

K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasi-Conformal Mappings in the Plane,, volume 48 of Princeton Mathematical Series. Princeton University Press, (2009). Google Scholar

[2]

R. Beals and R. R. Coifman, Linear spectral problems, nonlinear equations and the $\overline\partial$-method,, Inverse Problems, 5 (1989), 87. doi: 10.1088/0266-5611/5/2/002. Google Scholar

[3]

M. Boiti, J. Leon, M. Manna and F. Pempinelli, On a spectral transform of a Korteweg-de Vries equation in two spatial dimensions,, Inverse Problems, 2 (1986), 271. doi: 10.1088/0266-5611/3/1/008. Google Scholar

[4]

R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions,, Communications in Partial Differential Equations, 22 (1997), 1009. doi: 10.1080/03605309708821292. Google Scholar

[5]

R. Croke, J. Mueller, M. Music, P. Perry, S. Siltanen and A. Stahel, The Novikov-Veselov Equation: Theory and Computation,, , (). Google Scholar

[6]

L. D. Faddeev, Increasing solutions of the Schrödinger equation,, Soviet Physics Doklady, 11 (1966), 209. Google Scholar

[7]

P. G. Grinevich and R. G. Novikov, Faddeev eigenfunctions for point potentials in two dimensions,, Phys. Lett. A, 376 (2012), 1102. doi: 10.1016/j.physleta.2012.02.025. Google Scholar

[8]

M. Lassas, J. L. Mueller and S. Siltanen, Mapping properties of the nonlinear Fourier transform in dimension two,, Communications in Partial Differential Equations, 32 (2007), 591. doi: 10.1080/03605300500530412. Google Scholar

[9]

M. Lassas, J. L. Mueller, S. Siltanen and A. Stahel, The Novikov-Veselov equation and the inverse scattering method, Part I: Analysis,, Physica D: Nonlinear Phenomena, 241 (2012), 1322. doi: 10.1016/j.physd.2012.04.010. Google Scholar

[10]

M. Murata, Structure of positive solutions to $(-\Delta+V)u=0$ in $R^n$,, Duke Math. J., 53 (1986), 869. doi: 10.1215/S0012-7094-86-05347-0. Google Scholar

[11]

M. Music, P. Perry and S. Siltanen, Exceptional circles of radial potentials,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/4/045004. Google Scholar

[12]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math. (2), 143 (1996), 71. doi: 10.2307/2118653. Google Scholar

[13]

P. Perry, Miura maps and inverse scattering for the Novikov-Veselov equation,, Analysis and Partial Differential Equations, 7 (2014), 311. doi: 10.2140/apde.2014.7.311. Google Scholar

[14]

S. Siltanen, Electrical impedance tomography and Faddeev's Green functions,, Ann. Acad. Sci. Fenn. Mathematica Dissertationes, 121 (1999). Google Scholar

[15]

T.-Y. Tsai, The associated evolution equations of the Schödinger operator in the plane,, Inverse Problems, 10 (1994), 1419. doi: 10.1088/0266-5611/10/6/015. Google Scholar

[16]

T.-Y. Tsai, The Schrödinger operator in the plane,, Inverse Problems, 9 (1993), 763. doi: 10.1088/0266-5611/9/6/012. Google Scholar

show all references

References:
[1]

K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasi-Conformal Mappings in the Plane,, volume 48 of Princeton Mathematical Series. Princeton University Press, (2009). Google Scholar

[2]

R. Beals and R. R. Coifman, Linear spectral problems, nonlinear equations and the $\overline\partial$-method,, Inverse Problems, 5 (1989), 87. doi: 10.1088/0266-5611/5/2/002. Google Scholar

[3]

M. Boiti, J. Leon, M. Manna and F. Pempinelli, On a spectral transform of a Korteweg-de Vries equation in two spatial dimensions,, Inverse Problems, 2 (1986), 271. doi: 10.1088/0266-5611/3/1/008. Google Scholar

[4]

R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions,, Communications in Partial Differential Equations, 22 (1997), 1009. doi: 10.1080/03605309708821292. Google Scholar

[5]

R. Croke, J. Mueller, M. Music, P. Perry, S. Siltanen and A. Stahel, The Novikov-Veselov Equation: Theory and Computation,, , (). Google Scholar

[6]

L. D. Faddeev, Increasing solutions of the Schrödinger equation,, Soviet Physics Doklady, 11 (1966), 209. Google Scholar

[7]

P. G. Grinevich and R. G. Novikov, Faddeev eigenfunctions for point potentials in two dimensions,, Phys. Lett. A, 376 (2012), 1102. doi: 10.1016/j.physleta.2012.02.025. Google Scholar

[8]

M. Lassas, J. L. Mueller and S. Siltanen, Mapping properties of the nonlinear Fourier transform in dimension two,, Communications in Partial Differential Equations, 32 (2007), 591. doi: 10.1080/03605300500530412. Google Scholar

[9]

M. Lassas, J. L. Mueller, S. Siltanen and A. Stahel, The Novikov-Veselov equation and the inverse scattering method, Part I: Analysis,, Physica D: Nonlinear Phenomena, 241 (2012), 1322. doi: 10.1016/j.physd.2012.04.010. Google Scholar

[10]

M. Murata, Structure of positive solutions to $(-\Delta+V)u=0$ in $R^n$,, Duke Math. J., 53 (1986), 869. doi: 10.1215/S0012-7094-86-05347-0. Google Scholar

[11]

M. Music, P. Perry and S. Siltanen, Exceptional circles of radial potentials,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/4/045004. Google Scholar

[12]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math. (2), 143 (1996), 71. doi: 10.2307/2118653. Google Scholar

[13]

P. Perry, Miura maps and inverse scattering for the Novikov-Veselov equation,, Analysis and Partial Differential Equations, 7 (2014), 311. doi: 10.2140/apde.2014.7.311. Google Scholar

[14]

S. Siltanen, Electrical impedance tomography and Faddeev's Green functions,, Ann. Acad. Sci. Fenn. Mathematica Dissertationes, 121 (1999). Google Scholar

[15]

T.-Y. Tsai, The associated evolution equations of the Schödinger operator in the plane,, Inverse Problems, 10 (1994), 1419. doi: 10.1088/0266-5611/10/6/015. Google Scholar

[16]

T.-Y. Tsai, The Schrödinger operator in the plane,, Inverse Problems, 9 (1993), 763. doi: 10.1088/0266-5611/9/6/012. Google Scholar

[1]

Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure & Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727
[2]

Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems & Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371
[3]

Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149
[4]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solution of the Novikov equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2865-2899. doi: 10.3934/dcdsb.2018290
[5]

Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013
[6]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[7]

Priscila Leal da Silva, Igor Leite Freire. An equation unifying both Camassa-Holm and Novikov equations. Conference Publications, 2015, 2015 (special) : 304-311. doi: 10.3934/proc.2015.0304
[8]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803
[9]

Vladimir Georgiev, Atanas Stefanov, Mirko Tarulli. Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 771-786. doi: 10.3934/dcds.2007.17.771
[10]

Anatoli Babin, Alexander Figotin. Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1685-1718. doi: 10.3934/cpaa.2014.13.1685
[11]

Maike Schulte, Anton Arnold. Discrete transparent boundary conditions for the Schrodinger equation -- a compact higher order scheme. Kinetic & Related Models, 2008, 1 (1) : 101-125. doi: 10.3934/krm.2008.1.101
[12]

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
[13]

Juan-Ming Yuan, Jiahong Wu. The complex KdV equation with or without dissipation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 489-512. doi: 10.3934/dcdsb.2005.5.489
[14]

Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349
[15]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[16]

Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007
[17]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
[18]

Jongmin Han, Masoud Yari. Dynamic bifurcation of the complex Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 875-891. doi: 10.3934/dcdsb.2009.11.875
[19]

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159
[20]

Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences