Nonlinear Interactions and Dynamics in Problems From Fluids and Optics

本项目旨在从非线性光学、水波、多体量子系统和等离子体物理等方面研究数学物理中与哈密顿模型相关的拟线性色散偏微分方程的理论和计算。我们感兴趣的方程来自Schrödinger、Dirac、Korteweg-de Vries和重力-毛细管波方程。此外，PI希望继续探索紧域上哈密顿模型中与频率级联有关的强非线性效应。项目的大部分将集中在理论上理解强非线性模型解的存在性和规律性，以及研究这些复杂模型中特殊解的局部和全局动力学。然而，一个组成部分也将包括使用现代泛函分析技术来研究数值方法的收敛性和研究这些模型的离散近似的有效性。这样的分析可以作为动力学的动机，以及测试渐近极限的方法，在直接分析可能不再预测精确的动力学。此外，PI将致力于开发一个泛函分析框架，用于研究拟线性模型和计算近似的随机扰动下的稳定性。拟线性偏微分方程出现在曲率对底层物理有强烈影响的模型中。因此，流体中的表面张力、晶体中的表面能或光学中的相对论效应可以引入强烈依赖于溶液的高阶导数的相互作用项。这个项目的目的是阐明这些模型在什么意义上具有解，特别是在可能的情况下理解解的精确渐近描述。毛细波可能用于测量内部水波的表面特征，晶体弛豫在半导体制造中起着重要作用，超短脉冲激光器在最近的许多物理应用和等离子体产生中出现。从人力资源的角度来看，拟线性问题提供了大量的问题，可以用于培训从本科生到博士后的各级研究人员。特别是，本科生可以通过数值分析简单的1,2维玩具模型系统来学习拟线性方程的一些固有困难，研究生可以通过简化模型或临界方程的应用来学习泛函分析框架并开发技术，博士后可以通过项目合作来充分探索模型，开发技术并努力在稳定性理论和现象学中获得最佳结果。这些方法为受训者和PI提供了广泛的分析和计算工具，用于解决许多有趣的问题，使他们能够充分理解复杂的非线性逆问题和大量开放的非线性散射问题，以便在他们的职业生涯中很好地工作。

The goal of this project will be to study theoretically and computationally quasilinear dispersive partial differential equations related to Hamiltonian models in mathematical physics from nonlinear optics, water waves, many body quantum systems and plasma physics. The equations of interest come from families of Schrödinger, Dirac, Korteweg-de Vries, and gravity-capillary wave equations. In addition, the PI hopes to continue to explore the strongly nonlinear effects relating to frequency cascades in Hamiltonian models on compact domains. The bulk of the project will focus on understanding theoretically the existence and regularity of solutions to strongly nonlinear models, as well as studying local and global dynamics of special solutions within these complex models. However, a component will also consist of using modern functional analytic techniques to study convergence of numerical methods and to study validity of discrete approximations to these models. Such analysis can serve as motivation for dynamics, as well as a means of testing asymptotic limits where direct analysis may no longer predict precise dynamics. In addition, the PI will work to develop a functional analytic framework for studying stability under stochastic perturbations of quasilinear models and computational approximations. Quasilinear partial differential equations arise in models where curvature has a strong influence on the underlying physics. Hence, surface tension in fluids, surface energy in crystals or relativistic effects in optics can introduce interaction terms that strongly depend upon higher order derivatives of the solution. It is the aim of this project to shed light on in what sense these models have solutions and in particular understand precise asymptotic descriptions of the solutions when possible. Capillary waves might be useful for measuring the surface signature of internal water waves, crystal relaxation plays a role in semiconductor fabrication and ultra-short pulse lasers have appeared in many recent physics applications and plasma generation. From a human resources standpoint, quasilinear problems provide a large pool of problems that can be used for training purposes in work with researchers of all levels, from undergraduate to postdoctoral. In particular, undergraduates can learn some of the innate difficulties in quasilinear equations by numerically analyzing simple 1,2-dimensional toy-model systems, graduate students can learn the functional analysis framework and develop the techniques through applications to reduced models or critical equations, and postdocs can collaborate on projects to fully explore the models, develop techniques and work towards optimal results in stability theory and phenomenology. Such approaches give trainees and the PI a broad class of analytic and computational tools to use for solving many interesting problems, allowing them to work towards a full enough understanding to consider complex nonlinear inverse problems and a large set of open nonlinear scattering problems to work on well into their careers.