Two-dimensional inverse scattering for quasi-linear biharmonic operator

Abstract

<p style='text-indent:20px;'>The subject of this work concerns the classical direct and inverse scattering problems for quasi-linear perturbations of the two-dimensional biharmonic operator. The quasi-linear perturbations of the first and zero order might be complex-valued and singular. We show the existence of the scattering solutions to the direct scattering problem in the Sobolev space <inline-formula><tex-math id="M1">\begin{document}$ W^1_{\infty}( \mathbb{{R}}^2) $\end{document}</tex-math></inline-formula>. Then the inverse scattering problem can be formulated as follows: does the knowledge of the far field pattern uniquely determine the unknown coefficients for given differential operator? It turns out that the answer to this classical question is affirmative for quasi-linear perturbations of the biharmonic operator. Moreover, we present a numerical method for the reconstruction of unknown coefficients, which from the practical point of view can be thought of as recovery of the coefficients from fixed energy measurements.</p>