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 $(p-1)(n-1)>4$ and $k>(n-1)/2$, for every $f\in H^k(S_{\omega }^{n-1})$ of sufficiently small norm, we show there is a nonlinear Helmholtz eigenfunction taking the form \begin{equation*} u(r, ω) = r^{-(n-1)/2} \Big( e^{-iλr} f(ω) + e^{+iλr} b(ω) + O(r^{-ε}) \Big), \qquad \text{as } r \to \infty, \end{equation*} for some $b\in H^k(S_{\omega }^{n-1})$ and $\epsilon >0$. That is, the scattering matrix $f\mapsto b$ preserves Sobolev regularity, which is an improvement over the authors' previous work with Zhang, that proved a similar result with a loss of four derivatives.

24 pages, 1 figure