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Note on Calderón's inverse problem for measurable conductivities
February
2019, 13(1): 159-175.
doi: 10.3934/ipi.2019009
Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line
Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland |
We consider an inverse scattering problem of recovering the unknown coefficients of quasi-linearly perturbed biharmonic operator on the line. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove that the inverse Born approximation can be used to recover some essential information about the unknown coefficients from the knowledge of the reflection coefficient. This information is the jump discontinuities and the local singularities of the coefficients.
Keywords:
Inverse scattering,
biharmonic operator,
nonlinear perturbation,
Born approximation,
reconstruction of singularities.
Mathematics Subject Classification:
Primary: 34L25, 34A55; Secondary: 34L30.
Citation:
Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems and Imaging,
2019, 13
(1)
: 159-175.
doi: 10.3934/ipi.2019009
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References:
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F. Gazzola, H.-C. Grunau and G. Sweers,
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L. Hörmander,
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B. Pausader, Scattering for the beam equation in low dimensions, preprint,
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V. Serov, An inverse Born approximation for the general nonlinear Schrödinger operator on the line,
Journal of Physics A: Mathematical and General, 42 (2009), 332002, 7pp.
doi: 10.1088/1751-8113/42/33/332002. |
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V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D,
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doi: 10.1063/1.4769825. |
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E. C. Titchmarsh,
Introduction to the Theory of Fourier Integrals, Third edition. Chelsea Publishing Co., New York, 1986. |
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T. Tyni and M. Harju,
Inverse backscattering problem for perturbations of biharmonic operator,
Inverse Problems, 33 (2017), 105002, 20pp.
doi: 10.1088/1361-6420/aa873e. |
| [12] |
T. Tyni, M. Harju and V. Serov,
Recovery of singularities in fourth-order operator on the line from limited data,
Inverse Problems, 32 (2016), 075001, 22pp.
doi: 10.1088/0266-5611/32/7/075001. |
| [13] |
T. Tyni and V. Serov,
Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Prob. Imag., 12 (2018), 205-227.
doi: 10.3934/ipi.2018008. |
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E. Zeidler,
Applied Functional Analysis, Springer-Verlag, New York, 1995. |
show all references
References:
| [1] |
T. Aktosun and V. G. Papanicolaou, Time evolution of the scattering data for a fourth-order linear differential operator,
Inverse Problems, 24 (2008), 055013, 14 pp.
doi: 10.1088/0266-5611/24/5/055013. |
| [2] |
F. Gazzola, H.-C. Grunau and G. Sweers,
Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin Heidelberg, 2010.
doi: 10.1007/978-3-642-12245-3. |
| [3] |
L. Hörmander,
The Analysis of Linear Partial Differential Operators: Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin Heidelberg, 2003.
doi: 10.1007/978-3-642-61497-2. |
| [4] |
K. Iwasaki,
Scattering theory for 4th order differential operators: Ⅰ, Japan. J. Math., 14 (1988), 1-57.
doi: 10.4099/math1924.14.1. |
| [5] |
K. Iwasaki,
Scattering theory for 4th order differential operators: Ⅱ, Japan. J. Math., 14 (1988), 59-96.
doi: 10.4099/math1924.14.1. |
| [6] |
R. Kanwal,
Generalized Functions: Theory and Technique, Academic Press, New York, 1983. |
| [7] |
B. Pausader, Scattering for the beam equation in low dimensions, preprint,
arXiv: 0903.3777v2. |
| [8] |
V. Serov, An inverse Born approximation for the general nonlinear Schrödinger operator on the line,
Journal of Physics A: Mathematical and General, 42 (2009), 332002, 7pp.
doi: 10.1088/1751-8113/42/33/332002. |
| [9] |
V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D,
Journal of mathematical physics, 53 (2012), 123522, 16pp.
doi: 10.1063/1.4769825. |
| [10] |
E. C. Titchmarsh,
Introduction to the Theory of Fourier Integrals, Third edition. Chelsea Publishing Co., New York, 1986. |
| [11] |
T. Tyni and M. Harju,
Inverse backscattering problem for perturbations of biharmonic operator,
Inverse Problems, 33 (2017), 105002, 20pp.
doi: 10.1088/1361-6420/aa873e. |
| [12] |
T. Tyni, M. Harju and V. Serov,
Recovery of singularities in fourth-order operator on the line from limited data,
Inverse Problems, 32 (2016), 075001, 22pp.
doi: 10.1088/0266-5611/32/7/075001. |
| [13] |
T. Tyni and V. Serov,
Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Prob. Imag., 12 (2018), 205-227.
doi: 10.3934/ipi.2018008. |
| [14] |
E. Zeidler,
Applied Functional Analysis, Springer-Verlag, New York, 1995. |
Figure 2.
Numerical reconstruction (red) and the unknown combination (black)
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