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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

Teemu Tyni , and  Valery Serov

Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland

* Corresponding author: Teemu Tyni

Received  February 2018 Revised  September 2018 Published  December 2018

Fund Project: This work was supported by the Academy of Finland (application number 250215, the Centre of Excellence in Inverse Problems Research (2014–2017) and application number 312123, the Centre of Excellence of Inverse Modelling and Imaging (2018–2025)).

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 & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

K. Iwasaki, Scattering theory for 4th order differential operators: Ⅰ, Japan. J. Math., 14 (1988), 1-57.  doi: 10.4099/math1924.14.1.  Google Scholar

[5]

K. Iwasaki, Scattering theory for 4th order differential operators: Ⅱ, Japan. J. Math., 14 (1988), 59-96.  doi: 10.4099/math1924.14.1.  Google Scholar

[6]

R. Kanwal, Generalized Functions: Theory and Technique, Academic Press, New York, 1983.  Google Scholar

[7]

B. Pausader, Scattering for the beam equation in low dimensions, preprint, arXiv: 0903.3777v2. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Third edition. Chelsea Publishing Co., New York, 1986.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

K. Iwasaki, Scattering theory for 4th order differential operators: Ⅰ, Japan. J. Math., 14 (1988), 1-57.  doi: 10.4099/math1924.14.1.  Google Scholar

[5]

K. Iwasaki, Scattering theory for 4th order differential operators: Ⅱ, Japan. J. Math., 14 (1988), 59-96.  doi: 10.4099/math1924.14.1.  Google Scholar

[6]

R. Kanwal, Generalized Functions: Theory and Technique, Academic Press, New York, 1983.  Google Scholar

[7]

B. Pausader, Scattering for the beam equation in low dimensions, preprint, arXiv: 0903.3777v2. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Third edition. Chelsea Publishing Co., New York, 1986.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.  Google Scholar

Figure 1.  Coefficients $q_1$ and $q_0$
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Figure 2.  Numerical reconstruction (red) and the unknown combination (black)
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