Symmetries and Differential Equations

挪威数学家索菲斯·李开创了对称和微分方程组(DES)的学科，目的是把解决DES，特别是常微分方程组(ODES)的各种特殊技术的大杂烩理顺。他基本上表明，几乎所有已知的技术都可以通过对称性分析来系统化。现代求解常微分方程组的软件主要基于他的思想。*常微分方程组是描述一族曲线族(常微分方程组的解)的紧凑方程；偏微分方程式(PDE)是描述一族曲面族(其解)的紧凑方程。人们可以在不知道其解的情况下找到DE的对称性。*在包括工程、科学和经济在内的几乎所有应用领域的建模中都出现了DES。反过来，DES和/或其性质的解提供了基本的预测和解释。用解析和数值方法求解微分方程式的软件也在不断更新。DES的分析方法是基于找到对称性和/或守恒定律，然后使用对称性/CLS来寻找所提出的数据的解，或者如果这是不可能的，则寻找相关数据的解来检查数值方法的准确性。在求解微分方程组的数值方法中，最近的一些进展包含了对称性和/或守恒律的保持。正常情况下，不可能保持所有的对称性和/或CLS。这里一个重要的未来研究领域是确定对于给定的数据应该保留哪些对称性和/或CLS。*我在对称性和DES领域已经工作了大约50年，我在这个领域的工作导致了四本被广泛引用的研究书籍，其中包括一名前博士生和三名前PDF。这些都是由施普林格在其应用数学系列中出版的。我的工作大大扩展了Lie的工作，使其能够发现、计算和使用DES的对称性，从而可以求解更多的DES。这些扩展被广泛引用，并在中国、俄罗斯、美国和加拿大开发了许多新的软件来实现这些扩展。我经常受到许多国家的邀请，谈论我用东道国提供的资金开展的工作。基本上，我从NSERC获得的所有资金都用于支持学生和PDF。*在这项建议中，目的是进一步扩展对称方法，以便可以解决或简化应用程序中出现的更多DES。特别是，我的工作涉及为计算对称性开发新的空间。在我的所有活动中，我培训了许多研究生和PDF--我以前的很多研究生/PDF都在加拿大的行业工作过，包括量子计算、医疗诊断技术开发、航空时刻表、卫星成像和风险分析。**

The Norwegian mathematician Sophus Lie pioneered the subject of symmetries and differential equations (DEs) with the aim of putting order to the hodgepodge of ad-hoc techniques for solving DEs, especially ordinary differential equations (ODEs). He essentially showed that almost all known techniques could be systematized through symmetry analysis. Modern software for solving ODEs are based largely on his ideas.***An ODE is a compact equation describing a family of curves (the solutions of the ODE); a partial differential equation (PDE) is a compact equation for describing a family of surfaces (its solutions). One can find the symmetries of a DE without knowing its solutions. ***DEs arise in modelling in almost all applied fields, including engineering, science and economics. In turn, solutions of the DEs and/or their properties provide essential predictions and interpretations. Software continues to be updated for solving differential equations both analytically and numerically. Analytical methods for DEs are based on finding symmetries and/or conservation laws and then using symmetries/CLs to find solutions for posed data or, if this is not possible, to find solutions for related data to check accuracy of numerical methods. Some recent developments in numerical methods for solving DEs incorporate the preservation of symmetries and/or conservation laws. Normally, it is not possible to preserve all symmetries and/or CLs. Here an important future research area is to determine which symmetries and/or CLs should be preserved for given data.***I have worked in the field of symmetries and DEs for about 50 years and my work in this field has resulted in four well-cited research books with co-authors that have included a former PhD student and three former PDFs. All have been published by Springer in its Applied Math series. My work has significantly extended the work of Lie in being able to discover, calculate and use symmetries of DEs so that more DEs can be solved. These extensions have been well-cited and much new software has been developed to implement these extensions in China, Russia, USA and Canada. I am regularly invited by many countries to talk about my work with funding provided by the host country. Essentially all funding from my NSERC grant is used for supporting students and PDFs.***In this proposal, the aim is to further extend symmetry methods so that more DEs arising in applications can be solved or simplified. In particular, my work involves developing new spaces for computing symmetries. In all of my activities I train many graduate students and PDFs--a large number of my former grad students/PDFs have worked in Canadian industry, including quantum computing, development of medical diagnostic techniques, airline scheduling, satellite imaging, and risk analysis. **