  We study the direct and inverse scattering problem for the semilinear
Schr\"{o}dinger equation $\Delta u+a(x,u)+k^2u=0$ in $\mathbb{R}^d$. We show
well-posedness in the direct problem for small solutions based on the Banach
fixed point theorem, and the solution has the certain asymptotic behavior at
infinity. We also show the inverse problem that the semilinear function
$a(x,z)$ is uniquely determined from the scattering data. The idea is certain
linearization that by using sources with several parameters we differentiate
the nonlinear equation with respect to these parameter in order to get the
linear one.
