论文标题
麦克斯韦方程中的接口波包的严格信封近似，并具有2D定位
Rigorous Envelope Approximation for Interface Wave-Packets in Maxwell's Equations with 2D Localization
论文作者
论文摘要
我们在两个不均匀介电介电的界面上研究了依赖时间的Kerr非线性麦克斯韦方程，并研究横向磁性（矢量值）的横向磁（矢量值）。所得模型是准线性的。该问题在界面的每一侧都解决，并且字段通过自然界面条件耦合。波包位于界面处，并沿切向方向传播。对于缓慢调制的包膜近似，非线性schrödinger方程正式得出作为包膜的振幅方程。我们严格地证明在相应渐近较大的时间间隔上Sobolev空间规范中的近似值是合理的。准线性麦克斯韦问题的适应性结果是基于[R.的局部理论的。 Schnaubelt Und M. Spitz，具有保守界面条件的准麦克斯韦方程的局部良好性，\ textit {commun。数学。 Sci。}，被接受，2022]，并使用所涉及的自举参数将其扩展到小数据的渐近时间间隔。
We study transverse magnetic (vector valued) wave-packets in the time dependent Kerr nonlinear Maxwell's equations at the interface of two inhomogeneous dielectrics with an instantaneous material response. The resulting model is quasilinear. The problem is solved on each side of the interface and the fields are coupled via natural interface conditions. The wave-packet is localized at the interface and propagates in the tangential direction. For a slowly modulated envelope approximation the nonlinear Schrödinger equation is formally derived as an amplitude equation for the envelope. We rigorously justify the approximation in a Sobolev space norm on the corresponding asymptotically large time intervals. The well-posedness result for the quasilinear Maxwell problem builds on the local theory of [R. Schnaubelt und M. Spitz, Local wellposedness of quasilinear Maxwell equations with conservative interface conditions, \textit{Commun. Math. Sci.}, accepted, 2022] and extends this to asymptotically large time intervals for small data using an involved bootstrapping argument.
