On inverse scattering for the twodimensional nonlinear KleinGordon equation
Abstract
The inverse scattering problem for the twodimensional nonlinear KleinGordon equation $u_{tt}\Delta u + u = \mathcal{N}(u)$ is studied. We assume that the unknown nonlinearity $\mathcal{N}$ of the equation satisfies $\mathcal{N}\in C^\infty(\mathbb{R};\mathbb{R})$, $\mathcal{N}^{(k)}(y)=O(y^{\max\{ 3k,0 \}})$ ($y \to 0$) 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2024
 DOI:
 10.48550/arXiv.2406.06362
 arXiv:
 arXiv:2406.06362
 Bibcode:
 2024arXiv240606362S
 Keywords:

 Mathematics  Analysis of PDEs;
 35P25;
 35R30;
 35G20