Symmetries, Conserved Integrals, Hamiltonian Flows, and Integrable Systems

对称性、守恒积分、哈密顿流和可积系统

• 批准号: RGPIN-2019-06902
• 负责人: Anco, Stephen
• 金额: $ 1.53万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758621
• 关键词: Symmetries; Conserved; Integrals; Hamiltonian; Flows

Symmetries, conserved integrals, Hamiltonian structures, and related aspects of PDEs have attracted much mathematical activity in recent years. At the same time, there has been a surge of geometric approaches to the study of integrable systems in applied mathematics and mathematical physics. My proposal builds significantly on these directions of research in my current grant. Many parts are well suited for MSc projects, PhD theses, and postdoctoral work, as well as contributions by undergrad research students. (1) Integrable systems and curve/surface flows Using geometric frame methods, I have previously derived multi-component integrable systems which are group-invariant generalizations of nonlinear Schrodinger (NLS), modified Korteveg de Vries (mKdV), sine-Gordon (SG) equations with a bi-Hamiltonian structure, arising from curve flows in Riemannian symmetric spaces and semisimple Lie groups. This work will be extended and applied to obtain new integrable equations involving quaternionic and octonionic variables, which will be a large advance in the theory of integrable systems. I also will generalize some other earlier work on surface flows in Euclidean space to symmetric spaces and Lie groups,, which will lead to new integrable systems in 2+1 dimensions. (2) Peakon equations Peakons are travelling waves that have a peaked exponential profile. They arise in nonlinear dispersive wave equations connected to water wave theory. I have begun to work on several aspects of peakon equations, which includes finding a new peakon system related to the NLS equation. I have also shown that a very general family of wave equations possesses multi peakon solutions. I will continue to work on these two topics to understand and look at interesting features of peakon interactions. (3) Conserved integrals in fluid flow An open problem is to determine all conserved integrals for the fundamental equations of fluid flow. In recent work, I settled this problem for two important types of conserved integrals for compressible fluid flow in n>1 dimensions. I also extended the results by finding new conserved integrals on advected surfaces, which gave interesting generalizations of helicity and circulation. I plan to obtain carry out similar work for fluid flow with free boundaries, and for compressible magnetohydrodynamics. (4) Symmetries and conservation laws of nonlinear PDEs Conservation laws are very important in the study of nonlinear PDEs. For PDEs that have a Lagrangian, all conservation laws can be obtained from symmetries through Noether's theorem, but this connection fails for PDEs without a Lagrangian. In previous work published in several papers and a co-authored book, I have developed an algorithmic method to find the conservation laws admitted by any given PDE whether or not it has a Lagrangian. I plan to continue to develop and apply this method to PDEs of importance in applied mathematics and physics.

近年来，对称性，保守的积分，汉密尔顿结构和相关方面吸引了许多数学活动。同时，在应用数学和数学物理学中，对整合系统的研究进行了多种几何方法。我的建议在我目前的赠款中大大建立在这些研究方向上。许多部分非常适合MSC项目，博士论文和博士后工作，以及本科生研究专业的贡献。 （1）使用几何框架方法的可集成系统和曲线/表面流，我先前衍生出多组分的可集成系统，它们是非线性schrodinger（NLS）的群体不变概括（NLS），修改了Korteveg de vries（MKDV），Sine-Gordon（SG），Senine-gordon（SG），并在Bi-Hamilton结构中，ari arie curn curn curn curve curve curve curn curve curve curn curve curd curn curn curn curve。半圣谎言群体。这项工作将扩展并应用于获得涉及Quaternionic和Octonionic变量的新的可集成方程，这将是可整合系统理论的巨大进步。 我还将在欧几里得空间中的表面流上概括一些其他早期的工作，以对称空间和谎言组，这将导致2+1个维度的新型集成系统。 （2）Pearmon方程Pearcon是具有峰值指数曲线的行进波。它们出现在与水波理论相关的非线性分散波方程中。 我已经开始在Pearmon方程的几个方面进行工作，其中包括找到与NLS方程相关的新Peamon系统。我还表明，一个非常通用的波动方程家族具有多峰溶液。我将继续研究这两个主题，以理解并查看Pearpon Itess互动的有趣功能。 （3）流体流中保守的积分一个开放的问题是确定流体流的基本方程的所有保守积分。在最近的工作中，我解决了这两种重要类型的保守积分，以在n> 1维中进行可压缩流体流动。我还通过在对流表面上找到新的保守积分来扩展结果，这给出了有趣的螺旋和循环概括。我计划通过自由边界进行类似的流体流量以及可压缩的磁流失动力学。 （4）非线性PDES保护定律的对称性和保护定律在非线性PDE的研究中非常重要。对于具有Lagrangian的PDE，可以通过Noether定理从对称性获得所有保护法，但是对于没有Lagrangian的PDE，这种联系失败了。在以前发表的几篇论文和合着书籍的书中，我开发了一种算法方法，以找到任何给定PDE所接受的保护法律，无论它是否具有Lagrangian。 我计划继续开发并将这种方法应用于应用数学和物理学的重要性。