Regularity of the Scattering Matrix for Nonlinear Helmholtz Eigenfunctions

TL;DR

This paper proves that for certain nonlinear Helmholtz equations on asymptotically Euclidean or conic manifolds, the scattering matrix preserves Sobolev regularity, improving previous results by reducing derivative loss.

Contribution

The authors establish the regularity-preserving property of the nonlinear scattering matrix for a broad class of nonlinear Helmholtz eigenfunctions, with less regularity loss than prior work.

Findings

Scattering matrix preserves Sobolev regularity for small data.

Improves previous results by reducing derivative loss from four to none.

Applicable to nonlinear Helmholtz equations on asymptotically Euclidean or conic manifolds.

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 $f \in H^k(S^{n-1}_\omega)$ of sufficiently small norm, we show there is a nonlinear Helmholtz eigenfunction taking the form \begin{equation*} u(r, \omega) = r^{-(n-1)/2} \Big( e^{-i\lambda r} f(\omega) + e^{+i\lambda r} b(\omega) + O(r^{-\epsilon}) \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.