High-frequency limit of the Maxwell-Landau-Lifshitz system in the diffractive optics regime

TL;DR

This paper analyzes the high-frequency behavior of the Maxwell-Landau-Lifshitz system in one dimension, demonstrating long-time approximation by cubic Schrödinger equations using nonlinear normal form methods.

Contribution

It extends the analysis of Maxwell-Landau-Lifshitz systems to longer timescales, showing the validity of Schrödinger approximations over $O(1/\varepsilon)$ durations.

Findings

WKB solutions constructed over long times $O(1/\varepsilon)$

Schrödinger equations describe leading terms of solutions

Normal form method ensures approximation remains close

Abstract

We study semilinear Maxwell-Landau-Lifshitz systems in one space dimension. For highly oscillatory and prepared initial data, we construct WKB approximate solutions over long times $O(1/\e)$. The leading terms of the WKB solutions solve cubic Schr\"odinger equations. We show that the nonlinear normal form method of Joly, M\'etivier and Rauch applies to this context. This implies that the Schr\"odinger approximation stays close to the exact solution of Maxwell-Landau-Lifshitz over its existence time. In the context of Maxwell-Landau-Lifshitz, this extends the analysis of Colin and Lannes from times $O(|\ln \e|)$ up to $O(1/\e)$.