Recovery of singularities from fixed angle scattering data for biharmonic operator in dimensions two and three

The inverse fixed angle problem for operator ${\mathrm{\Delta }}^{2}u+V(x,\mathrm{\mid }u\mathrm{\mid })u$ is considered in dimensions $n=2,3$. We prove that the difference between an inverse fixed angle Born approximation and the function $V(\cdot ,1)$ is smoother than the function $V$ itself in some Sobolev scale. This allows us to conclude that the main singularities of the perturbation $V$ can be reconstructed from the knowledge of the scattering amplitude with some fixed incident angle.