Title: A Hamiltonian structure-preserving discretization of Maxwell's equations in nonlinear media Show Chinese title

Title: 非线性介质中保Hamilton结构的Maxwell方程离散化

Abstract: A simple Hamiltonian modeling framework for general models in nonlinear optics is given. This framework is specialized to describe the Hamiltonian structure of electromagnetic phenomena in cubicly nonlinear optical media. The model has a simple Poisson bracket structure with the Hamiltonian encoding all of the nonlinear coupling of the fields. The field-independence of the Poisson bracket facilitates a straightforward Hamiltonian structure-preserving discretization using finite element exterior calculus. The generality and relative simplicity of this Hamiltonian framework makes it amenable for simulating a broad class of time-domain nonlinear optical problems. The main contribution of this work is a finite element discretization of Maxwell's equations in cubicly nonlinear media which is energy-stable and exactly conserves Gauss's laws. Moreover, this approach may be readily adapted to consider more general nonlinear media in subsequent work. Show Chinese abstract

Abstract: 给出了一个用于非线性光学中一般模型的简单哈密顿建模框架。 该框架被专门用于描述立方非线性光学介质中电磁现象的哈密顿结构。 该模型具有简单的泊松括号结构，其中哈密顿量编码了场的所有非线性耦合。 泊松括号对场的独立性使得利用有限元外微积分可以方便地实现保持哈密顿结构的离散化。 这种哈密顿框架的通用性和相对简单性使其适用于模拟广泛的时域非线性光学问题。 本文的主要贡献是在立方非线性介质中对麦克斯韦方程组进行了一种能量稳定的有限元离散化，并精确保持了高斯定律。 此外，这种方法可以很容易地推广到后续工作中更一般的非线性介质。