On periodic solutions and attractors for the Maxwell--Bloch equations

Alexander Komech
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Abstract

We consider the Maxwell-Bloch system which is a finite-dimensional approximation of the coupled nonlinear Maxwell-Schr\"odinger equations. The approximation consists of one-mode Maxwell field coupled to two-level molecule. We construct time-periodic solutions to the factordynamics which is due to the symmetry gauge group. For the corresponding solutions to the Maxwell--Bloch system, the Maxwell field, current and inversion are time-periodic, while the wave function acquires a unit factor in the period. The proofs rely on high-amplitude asymptotics of the Maxwell field and a development of suitable methods of differential topology: the transversality and orientation arguments. We also prove the existence of the global compact attractor.
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论麦克斯韦-布洛赫方程的周期解和吸引子
我们考虑麦克斯韦-布洛赫系统,它是耦合非线性麦克斯韦-薛定谔方程的有限维近似。近似由单模麦克斯韦场耦合到两能级分子组成。构造了由对称规群引起的因子动力学的时间周期解。对于Maxwell—Blochsystem的对应解,Maxwell场、电流和反演都是时间周期的,而波函数在周期内获得一个单位因子。这些证明依赖于麦克斯韦场的高振幅渐近性和微分拓扑的合适方法的发展:横向和方向论证。我们还证明了全局紧吸引子的存在性。
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