On periodic solutions for the Maxwell–Bloch equations

Abstract

We consider the Maxwell–Bloch system which is a finite-dimensional approximation of the coupled nonlinear Maxwell–Schrödinger equations. The approximation consists of one-mode Maxwell field coupled to $N\ge 1$ two-level molecules. Our main result is the existence of solutions with time-periodic Maxwell field. For the proof we construct time-periodic solutions to the reduced system with respect to the symmetry gauge group $U\left(1\right)$. The solutions correspond to fixed points of the Poincaré map, which are constructed using the contraction of high-amplitude Maxwell field and the Lefschetz theorem. The theorem is applied to a suitable modification of the reduced equations which defines a smooth dynamics on the compactified phase space. The crucial role is played by the fact that the Euler characteristic of the compactified space is strictly greater than the same of the infinite component.