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

Abstract

The inverse scattering problem for the two-dimensional nonlinear Klein-Gordon equation $u_{tt}-\Delta u+u=\mathcal{N}(u)$ is studied. We assume that the unknown nonlinearity $\mathcal{N}$ of the equation satisfies $\mathcal{N}\in C^{\infty }(\mathbb{R};\mathbb{R})$, ${\mathcal{N}}^{(k)}(y)=O(|y{|}^{\max \{3-k,0\}})$ ($y\to 0$) and ${\mathcal{N}}^{(k)}(y)=O(e^{c{y}^2})$ ($|y|\to \infty$) 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.

24 pages. Final version published in J. Hyperbolic Differ. Equ