Applications of advances in computer algebra to studying classical integrable systems and related algebraic structures

应用计算机代数的进展来研究经典可积系统和相关代数结构

• 批准号: RGPIN-2017-06330
• 负责人: Wolf, Thomas
• 金额: $ 1.75万
• 依托单位: Brock University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679551
• 关键词: Applications; advances; computer; algebra; studying

The area of my work is the study of differential equations and the development of computer algebra algorithms and programs to investigate these equations. Partial differential equations (PDEs) describe the behaviour of a quantity in space and time. They are especially interesting if they allow solutions that propagate and keep their shape or solutions that have shape preserving structures propagating towards each other, penetrating each other and afterwards taking again their original shape. Such PDEs have a number of unusual properties, for example, they have infinitely many conserved quantities and infinitely many infinitesimal symmetries. They are called 'integrable'. To find these integrable PDEs one formulates mathematical conditions for their special properties in the form of auxiliary equations and tries to solve them. These conditions have in common that they are overdetermined, i.e. they involve more conditions than unknowns. The idea is to have one powerful package of computer programs to solve overdetermined systems and to use that repeatedly to find integrable differential equations of different kind.***Work under this grant application has two aims: the further strengthening of the computer algebra package Crack for solving overdetermined systems and its application to integrability problems:***- Inversion of Recursion and Hamiltonian Operators***- Classification of integrable evolution-type equations with fermionic variables***- Integrable ODEs with matrix variables***- Bi-hamiltonian structures and Poisson cohomologies of Elliptic Algebras***Some integrable PDEs do have a practical importance. One can manufacture glass fibers such that light propagation in these fibers is described by the integrable Schroedinger equation. This equation has solutions (propagating pulses of light) that keep their shape, and thus such cables are perfectly suited to transmitting large amounts of data over long distances.*****

我的工作领域是微分方程的研究以及计算机代数算法和程序的开发来研究这些方程。偏微分方程 (PDE) 描述了空间和时间上的量的行为。如果它们允许传播并保持其形状的解决方案或具有形状保持结构的解决方案相互传播、相互渗透并随后再次恢复其原始形状，那么它们就特别有趣。此类偏微分方程具有许多不寻常的性质，例如，它们具有无穷多个守恒量和无穷多个无穷小对称性。它们被称为“可积”。为了找到这些可积偏微分方程，我们需要以辅助方程的形式为其特殊性质制定数学条件，并尝试求解它们。这些条件的共同点是它们是多决定的，即它们涉及的条件多于未知数。这个想法是拥有一个强大的计算机程序包来解决超定系统，并重复使用它来找到不同类型的可积微分方程。***这项拨款申请下的工作有两个目标：进一步加强计算机代数包破解用于求解超定系统及其在可积性问题中的应用：***- 递归和哈密顿算子的反演***- 具有费米子变量的可积演化型方程的分类***- 具有矩阵变量的可积常微分方程***-椭圆代数的双哈密顿结构和泊松上同调***一些可积偏微分方程确实具有实际重要性。人们可以制造玻璃纤维，使得这些纤维中的光传播由可积薛定谔方程描述。该方程具有保持其形状的解（传播光脉冲），因此此类电缆非常适合长距离传输大量数据。*****