Reconstruction of singularities in two-dimensional quasi-linear biharmonic operator

Abstract

<p style='text-indent:20px;'>The inverse backscattering Born approximation for two-dimensional quasi-linear biharmonic operator is studied. We prove the precise formulae for the first nonlinear term of the Born sequence. We prove also that all other terms in this sequence are <inline-formula><tex-math id="M1">\begin{document}$ H^t- $\end{document}</tex-math></inline-formula>functions for any <inline-formula><tex-math id="M2">\begin{document}$ t&lt;1 $\end{document}</tex-math></inline-formula>. These formulae and estimates allow us to conclude that all main singularities of a certain combination of unknown coefficients, in particular, <inline-formula><tex-math id="M3">\begin{document}$ L^p- $\end{document}</tex-math></inline-formula>singularities for <inline-formula><tex-math id="M4">\begin{document}$ 2\le p&lt;\infty $\end{document}</tex-math></inline-formula>, can be uniquely reconstructed using the inverse backscattering Born approximation. In addition, it is shown that the jumps (<inline-formula><tex-math id="M5">\begin{document}$ L^{\infty}- $\end{document}</tex-math></inline-formula>singularities) over smooth curves are uniquely determined by the backscattering data and can be recovered from the Born approximation. We present a numerical method for the reconstruction of these singularities.</p>