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

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/ε). The leading terms of the WKB solutions solve cubic Schrödinger equations. We show that the nonlinear normal form method of Joly, Métivier and Rauch [J. Diff. Eq. 166 (2000), 175–250] applies to this context. This implies that the Schrödinger approximation stays close to the exact solution of Maxwell–Landau–Lifshitz over diffractive times. In the context of Maxwell–Landau–Lifshitz, this extends the analysis of Colin and Lannes [Discrete and Continuous Dynamical Systems 11(1) (2004), 83–100] from times O(|ln ε|) up to O(1/ε).