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

The inverse scattering problem for the two-dimensional nonlinear Klein-Gordon equation uₜₜ-Δ u + u = N(u) is studied. We assume that the unknown nonlinearity N of the equation satisfies N∈ C^∞(R;R), N⁽ᵏ⁾(y)=O(|y|^max\ 3-k,0 \) (y → 0) and N⁽ᵏ⁾(y)=O(ec y²) (|y| → ∞) for any k=0,1,2,⋯. Here, c is a positive constant. We establish a reconstraction formula of N⁽ᵏ⁾(0) (k=3,4,5,⋯) by the knowledge of the scattering operator for the equation. As an application, we also give an expression for higher order G\ᵃteaux differentials of the scattering operator at 0.