Well-posedness and exponential stability of dispersive nonlinear Maxwell equations with PML: An evolutionary approach

Abstract

This paper presents a mathematical foundation for physical models in nonlinear optics through the lens of evolutionary equations. It focuses on two key concepts: well-posedness and exponential stability of Maxwell equations, with models that include materials with complex dielectric properties, dispersion, and discontinuities. We use a Hilbert space framework to address these complex physical models in nonlinear optics. While our focus is on the first-order formulation in space and time, higher solution regularity recovers and equates to the second-order formulation. We incorporate perfectly matched layers (PMLs), which model absorbing boundary conditions, to facilitate the development of numerical methods. We demonstrate that the combined system remains well-posed and exponentially stable. Our approach applies to a broad class of partial differential equations (PDEs) and accommodates materials with nonlocal behavior in space and time. The contribution of this work is a unified framework for analyzing wave interactions in advanced optical materials.