We consider the problem of the propagation of an electric field generated by periodic pumping in a stable medium of two-level atoms as the mixed problem for the Maxwell–Bloch equations without spectrum broadening. An approach to the study of such a problem is proposed. We use the inverse scattering transform method in the form of the matrix Riemann–Hilbert (RH) problem, using simultaneous spectral analysis of both the Lax equations. The proposed matrix RH problem solves the problem of the propagation of a sinusoidal signal in an unperturbed stable medium (attenuator). It is proved that this RH problem provides the causality principle for the region t < x, and for the region of the light cone, 0 < x < t allows us to find the asymptotics of the transmitted signal. First, we study the asymptotics of the RH problem for large times, and then, we obtain asymptotic formulas for the mixed problem solution of the Maxwell–Bloch equations when the attenuator is long enough. Three sectors are obtained in the light cone where the asymptotics have essentially different behaviors.

中文翻译：

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在麦克斯韦-布洛赫模型的两能级原子的稳定介质中周期性泵浦产生的电场的传播

我们将周期性泵浦在两能级原子的稳定介质中产生的电场的传播问题视为麦克斯韦-布洛赫方程的混合问题，而没有频谱展宽。提出了研究此类问题的方法。我们使用矩阵黎曼-希尔伯特 (RH) 问题形式的逆散射变换方法，同时使用两个 Lax 方程的光谱分析。提出的矩阵 RH 问题解决了正弦信号在未受扰动的稳定介质（衰减器）中的传播问题。证明这个RH问题为t < x区域提供了因果关系原理，对于光锥区域，0 < x < t可以让我们找到传输信号的渐近线。首先，我们大量研究 RH 问题的渐近性，然后，当衰减器足够长时，我们获得了麦克斯韦-布洛赫方程的混合问题解的渐近公式。在光锥中获得三个扇区，其中渐近线具有本质上不同的行为。