论文标题
Maxwell-Bloch方程的初始符合价值问题，具有任意不均匀扩展和周期性边界功能
Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function
论文作者
论文摘要
研究了麦克斯韦布洛克方程的初始有限值问题（IBVP），并研究了任意不均匀扩展和周期性边界条件。该IBVP描述了通过在具有分布式两级原子的谐振介质中定期抽水产生的电磁波的传播。我们以矩阵riemann-hilbert问题的形式扩展了反向散射变换方法，用于求解所考虑的IBVP。使用ablowitz-kaup-newell-segur方程的系统等效于麦克斯韦 - 布洛赫（MB）方程的系统，我们构建了相关的矩阵riemann-hilbert（rh）问题。证明了关于构建的RH问题解决方案的存在，独特性和平滑性特性的定理，并证明了所考虑的IBVP的解决方案是由相关RH问题的解决方案唯一定义的。事实证明，RH问题提供了因果关系原则。给出了MB方程解决方案在相关RH问题的解决方案方面的表示。该方法的重要性还在于，在研究了构建的RH问题和等效问题的渐近行为后，我们可以为MB方程的相应IBVP解决方案的渐近剂提供公式。
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.
