The smoothness of scattering operators for sinh-Gordon and nonlinear Schrödinger equations
The author proves the existence and the infinite smoothness of the scattering operator for the sinh-Gordon equation \[ u_{tt}-\Delta u +\frac{(e^u-e^{-u})}{2}=0. \] More precisely, one proves that for \(n\geq 3\), \(s\geq n/2\) there exists \(d>0\) such that the scattering operator carries the set \(\{(\varphi, \psi)\in H^s\times H^{s-1} \mid \|\varphi\|_{H^n/2}+ \|\psi\|_{H^{n/2-1}}\leq \delta\}\) into \(H^s\times H^{s-1}\). The proof is based on some nonlinear estimates in Besov spaces. It is also stated that a similar result holds for the nonlinear Schrödinger equation \[ i u_t+\Delta u +\mu (e^{\lambda |u|^2}-1)u=0. \]
scientific article; zbMATH DE number 1885855