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。自从具有高峰功率的强激光问世以来，这一领域已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组的大量模拟文献相比，开发非线性光学介质中时空模型的数学理解计算工具相对较少。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，从而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麦克斯韦模型的理解和模拟方面取得重大进展，其目的是：（1）为非线性光学社区提供强大的模拟工具，（2）开发专门针对不同类型的非线性模型的新型数学和数值技术。具体的技术方面包括开发能量稳定的时间离散化以及两类空间离散化，不连续Galerkin方法和模拟有限差分方法，用于非线性（色散）光学介质中电磁波的传播。宏观唯象和微观量子描述将被视为建模的非线性材料的响应。飞秒孤子传播，谐波产生，自聚焦等应用将被模拟，并与现有的时域方法进行比较。这个合作计划通过一个有凝聚力的研究计划得到加强，该计划依赖于每个主要研究者的互补专业知识。通过对研究生的培训，将教育部分结合起来。

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.