On inverse scattering for the two-dimensional nonlinear Klein-Gordon equation

The inverse scattering problem for the two-dimensional nonlinear Klein-Gordon equation $u_{tt}-\mathrm{\Delta }u+u=\mathcal{N}(u)$ is studied. We assume that the unknown nonlinearity $\mathcal{N}$ of the equation satisfies $\mathcal{N}\in C^{\mathrm{\infty }}(\mathbb{R};\mathbb{R})$, ${\mathcal{N}}^{(k)}(y)=O(\mathrm{\mid }y{\mathrm{\mid }}^{\max \{3-k,0\}})$ ($y\to 0$) and ${\mathcal{N}}^{(k)}(y)=O(e^{c{y}^2})$ ($\mathrm{\mid }y\mathrm{\mid }\to \mathrm{\infty }$) for any $k=0,1,2,\cdots$. Here, $c$ is a positive constant. We establish a reconstraction formula of ${\mathcal{N}}^{(k)}(0)$ ($k=3,4,5,\cdots$) by the knowledge of the scattering operator for the equation. As an application, we also give an expression for higher order Gâteaux differentials of the scattering operator at 0.