Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line

Teemu Tyni; Valery Serov

doi:10.3934/ipi.2019009

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2019, Volume 13, Issue 1: 159-175. Doi: 10.3934/ipi.2019009

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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
* Corresponding author: Teemu Tyni

Received: January 31, 2018

Revised: August 31, 2018

Published: December 2018

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)).

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Abstract

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.

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Mathematics Subject Classification: Primary: 34L25, 34A55; Secondary: 34L30.

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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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