具有任意非均匀展宽和周期边界函数的Maxwell-Bloch方程的初边值问题
Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function
AI总结：
本文研究含任意非均匀展宽与周期边界条件的Maxwell-Bloch方程初边值问题，通过等价方程组构建矩阵Riemann-Hilbert问题，证明解的相关性质，给出解的表示并可推导其渐近行为。
摘要：
本文研究了具有任意非均匀展宽和周期边界条件的Maxwell-Bloch方程的初边值问题（IBVP）。该初边值问题描述了在含有分布式二能级原子的共振介质中，由周期泵浦产生的电磁波的传播过程。我们拓展了矩阵Riemann-Hilbert问题形式的逆散射变换方法，用于求解所研究的初边值问题。利用与Maxwell-Bloch（MB）方程组等价的Ablowitz-Kaup-Newell-Segur方程组，我们构建了对应的矩阵Riemann-Hilbert（RH）问题。本文证明了所构建的RH问题解的存在性、唯一性和光滑性相关定理，并表明所研究的初边值问题的解可由对应的RH问题的解唯一确定。同时证明了该RH问题满足因果律。文中给出了MB方程组的解用对应的RH问题的解表示的形式。该方法的重要性还在于，通过研究所构建的RH问题及等价问题的渐近行为，我们可以得到MB方程对应初边值问题解的渐近公式。
英文摘要：
The initial-boundary value problem (IBVP) for the Maxwell-Bloch equations with an arbitrary inhomogeneous broadening and periodic boundary condition is studied. This IBVP describes the propagation of an electromagnetic wave generated by periodic pumping in a resonant medium with distributed two-level atoms. We extended the inverse scattering transform method in the form of the matrix Riemann-Hilbert problem for solving the considered IBVP. Using the system of Ablowitz-Kaup-Newell-Segur equations equivalent to the system of the Maxwell-Bloch (MB) equations, we construct the associated matrix Riemann-Hilbert (RH) problem. Theorems on the existence, uniqueness and smoothness properties of a solution of the constructed RH problem are proved, and it is shown that a solution of the considered IBVP is uniquely defined by the solution of the associated RH problem. It is proved that the RH problem provides the causality principle. The representation of a solution of the MB equations in terms of a solution of the associated RH problem are given. The significance of this method also lies in the fact that, having studied the asymptotic behavior of the constructed RH problem and equivalent ones, we can obtain formulas for the asymptotics of a solution of the corresponding IBVP for the MB equations.
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