OP: Collaborative Research: Compatible Discretizations for Maxwell Models in Nonlinear Optics

非线性光学是研究光在非线性介质中的行为的科学。自从高峰值功率的强激光问世以来，这一领域已经发展成为物理学的一个重要分支。与线性光学介质中麦克斯韦方程模拟的大量文献相比，为非线性光学介质中的时空模型开发数学上易于理解的计算工具相对较少受到计算数学界的关注。这方面的重大进展可以为科学界提供可靠和准确的工具来模拟和理解非线性光学现象，因此可以更好地用于实际应用。合作研究计划的目标是在非线性光学中对Maxwell模型的理解和模拟方面取得重大进展，目的是：(1)为非线性光学社区提供强大的模拟工具，(2)开发专门为不同类型的非线性模型量身定做的新型数学和数值技术。具体的技术方面包括发展能量稳定的时间离散以及两类空间离散：间断伽辽金方法和模拟有限差分方法，用于非线性(色散)光学介质中的电磁波传播。我们将同时考虑宏观唯象和微观量子描述来模拟材料的非线性响应。我们将对飞秒孤子传输、谐波产生、自聚焦等应用进行模拟，并与现有的时间域方法进行比较。这一协作计划通过依赖于每个主要研究人员的互补专业知识的连贯的研究计划而得到加强。通过对研究生的培训，将教育组成部分整合起来。

Nonlinear optics is the study of the behavior of light in nonlinear media. This field has developed into a significant branch of physics since the introduction of intense lasers with high peak powers. Compared with the huge amount of literature on simulations of Maxwell's equations in linear optical media, developing mathematically well-understood computational tools for space-time models in nonlinear optical media is relatively less tackled by the computational math community. Major advancement in this aspect can provide the scientific community reliable and accurate tools to simulate and to understand nonlinear optical phenomena, which hence can be better harnessed for practical applications. The objective of the collaborative research program is to make significant advances in the understanding and simulations of Maxwell models in nonlinear optics with the aim of: (1) providing robust simulation tools for the nonlinear optics community, (2) developing novel mathematical and numerical techniques that are specifically tailored for different types of nonlinear models. The specific technical aspect includes the development of energy-stable time discretizations as well as two classes of spatial discretizations, discontinuous Galerkin methods and mimetic finite difference methods, for the propagation of electromagnetic waves in nonlinear (dispersive) optical media. Both macroscopic phenomenological and microscopic quantum descriptions will be considered for modeling the nonlinear material responses. Applications involving femtosecond soliton propagation, harmonic generation, self focusing, among others will be simulated and compared to existing time domain methods. This collaborative program is strengthened by a cohesive research plan that relies on the complementary expertise of each principal investigator. The educational components are integrated through the training of graduate students.