Regularity of the scattering matrix for nonlinear Helmholtz eigenfunctions

Abstract

We study the nonlinear Helmholtz equation $(\Delta -{\lambda }^2)u=\pm |u{|}^{p-1}u$ on ${\mathbb{R}}^n$, $\lambda >0$, $p\in \mathbb{N}$ odd, and more generally $({\Delta }_g+V-{\lambda }^2)u=N[u]$, where ${\Delta }_g$ is the (positive) Laplace–Beltrami operator on an asymptotically Euclidean or conic manifold, $V$ is a short range potential, and $N[u]$ is a more general polynomial nonlinearity. Under the conditions $(p-1)(n-1)/2>2$ and $k>(n-1)/2$, for every $f\in H^k({\mathbb{S}}_{\omega }^{n-1})$ of sufficiently small norm, we show there is a nonlinear Helmholtz eigenfunction taking the form

for some $b\in H^k({\mathbb{S}}_{\omega }^{n-1})$ and $ϵ>0$. That is, the nonlinear scattering matrix $f\mapsto b$ preserves Sobolev regularity, which is an improvement over the authors' previous work (2020) with Zhang, that proved a similar result with a loss of four derivatives.