Applications of Lie Pseudogroups

本计画将开发新的计算工具，以研究非线性偏微分方程式的无穷维对称群。 该方法是基于适应的有限维等变移动框架法以前开发的主要研究者。 算法直接确定这种对称群的结构和他们的代数微分不变量，寻找明确的，非不变的解决方案，测试线性化和可积性，并确定守恒定律和可积性约束的变分法中出现的问题将开发。 具体的应用将包括边界层和其他流体模型，孤子在更高的空间维度，准地转系统在气象学。 在无限维李伪群的几何基础的进展也是预期的。 设计和应用通用计算机代数软件包来处理所需的计算将是该项目的一个组成部分。利用对称性和不变性仍然是研究和解决所有应用领域中出现的复杂非线性系统的最强大的通用数学工具之一。 该项目的重点将是开发和利用处理无限维对称群的新方法，这些对称群出现在广泛的应用中，包括流体和等离子体力学、磁流体力学、气象学和气候建模、基础物理学的相对论和规范理论、弹性和可积系统/孤子模型，因此，该项目的结果预计将对物理学、工程学和其他领域的非线性现象的求解方法和随后的理解产生重大影响。

This project will develop new computational tools for studying infinite-dimensional symmetry groups of nonlinear partial differential equations. The approach is based on an adaptation of the finite-dimensional equivariant moving frame method previously developed by the principal investigator. Algorithms for directly determining the structure of such symmetry groups and their algebra of differential invariants, for finding explicit, non-invariant solutions, for testing for linearizability and integrability, and for determining conservation laws and integrability constraints for problems arising in the calculus of variations will be developed. Specific applications will include boundary layer and other fluid models, solitons in higher space dimensions, and quasi-geostrophic systems in meteorology. Advances in the geometric foundations of infinite-dimensional Lie pseudo-groups are also anticipated. The design and application of general-purpose computer algebra packages for handling the required computations will be an integral part of the project.Exploitation of symmetry and invariance properties remains one of the most powerful general-purpose mathematical tools for studying and solving the complex nonlinear systems that arise in all areas of applications. The focus of this project will be to develop and exploit new methods for handling infinite-dimensional symmetry groups, which arise in a broad range of applications, including fluid and plasma mechanics, magneto-hydrodynamics, meteorology and climate modeling, relativity and gauge theories from fundamental physics, elasticity, and integrable systems/soliton models, and so on. The results of this project are thus anticipated to have a significant impact on the solution methods and consequent understanding of nonlinear phenomena in physics, engineering, and elsewhere.