Infinite Dimensional Geometric Singular Perturbation Theory for the Maxwell--Bloch Equations

Abstract

We study the Maxwell--Bloch equations governing a two-level laser in a ring cavity. For Class A lasers, these equations have two widely separated time scales and form a singularly perturbed, semilinear hyperbolic system with two distinct characteristics. We extend Fenichel's geometric singular perturbation theory [N. Fenichel, J. Differential Equations, 31 (1979), pp. 53--98] to the Maxwell--Bloch equations by proving the persistence of a $C^k, 0<k< \infty$, slow manifold under an unbounded perturbation. The proof is obtained by a modified graph transform method. We use uniform decay estimates of Constantin, Foias, and Gibbon [Nonlinearity, 2 (1989), pp. 241--269] to obtain a cone condition. These estimates rely on the energy preserving nature of the nonlinearity and the existence of two distinct characteristics. The cone condition and the fact that the unbounded perturbation generates a continuous group are used to define the graph transform. The slow manifold is a globally attracting, positively invariant manifold, with infinite dimension and codimension, that contains the attractor of the system. The slow manifold depends only continuously on $\varepsilon$ and converges uniformly on (strongly) compact sets to the critical manifold. This enables us to rigorously decouple the slow and fast time scales and obtain a reduced (but still infinite-dimensional) dynamical system described by a functional differential equation.