Regularity and Stability for Solutions of Quasilinear Wave Equations with Singularities

光学系统中的尖点波和冲击波、液晶和水波的形成是自然界和工程中出现的非线性现象的例子，可以用非线性偏微分方程来描述。该项目考虑了波动模型的非线性偏微分方程，其解可能在有限时间内形成不同类型的奇点，例如浅水波动方程中的尖点奇点或非线性光学模型中的冲击波，其总体目标是了解在何种条件下解始终有效。该研究将在工程上提供指导，例如设计和控制光学系统中的设备。该项目还将为研究生的研究培训提供机会。该项目的一个主要目标是描述非线性波速的幂如何影响解的规则性。研究人员将研究一类方程，其中包括来自非线性光学的短脉冲方程和来自水波的Camassa-Holm型方程，其解可能形成有限时间奇点。他们还将建立一个最佳的传输度量时，研究稳定性的波动方程系统建模的液晶。最后，他们将探索具有径向对称性的高维准线性模型，即所谓的O（3）sigma模型，其背景是广义相对论，该项目由DMS应用数学计划和刺激竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The formation of cusp and shock waves in optical systems, liquid crystals, and water waves are examples of nonlinear phenomena arising in nature and engineering that can be described by nonlinear partial differential equations. This project considers nonlinear partial differential equations for wave models whose solutions might form different types of singularities in finite time, such as cusp singularities in the shallow water wave equation or shock waves in models of nonlinear optics, with the overall aim of understanding under which condition the solutions are valid for all times. The research will give guidance in engineering, for example for designing and controlling devices in optical systems. The project will also provide opportunities for research training of graduate students. A main goal of the project is to describe how the power of nonlinear wave speed impacts the regularity of solutions. The investigator will study a class of equations, which include the short pulse equation from nonlinear optics and Camassa-Holm type equations from water waves, whose solutions might form finite-time singularities. They will also establish an optimal transport metric when studying the stability of a system of wave equations modelling nematic liquid crystals. Finally, they will explore a higher dimensional quasilinear model with radial symmetry, the so-called O(3) sigma-model, with background in general relativity, Yang-Mills field and nematic liquid crystals.This project is jointly funded by the DMS Applied Mathematics Program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.