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-2014-05787
• 负责人: Anco, Stephen
• 金额: $ 1.31万
• 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: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567390
• 关键词: Symmetries; conserved; integrals; Hamiltonian; flows

Symmetries, conserved integrals, Hamiltonian structures, and related aspects of PDEs have attracted much activity in recent years. At the same time, there has been a surge of geometric approaches to the study of integrable PDE systems. 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. Bi-Hamiltonian integrable systems and curve/surface flows: Using geometric frame methods, I recently derived universal 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. In work underway with MSc students, we are deriving new types of NLS/mKdV eqns. from Hermitian and Lorentzian symmetric spaces. I plan to derive new quaternion and matrix types of eqns.from quaternion and Grassmannian spaces. I also plan to apply similar methods to an important open problem of how to derive group-invariant integrable systems from curve flows in homogeneous spaces. I intend to extend my method to derive Lax pairs. This will lead to a geometric understanding of Drinfeld&Sokolov's universal construction of Lax pairs in affine Lie algebras, which is a central result in the theory of integrable systems. I also intend to lift the Lax pairs and bi-Hamiltonian structures from the frame variables back to the curve flow. This will give an explicit formulation of abstract results known on Lax pairs and Poisson brackets. Generalizing work I have done in R^3, I plan to derive 2+1 dimensional integrable systems from geometric surface flows in symmetric spaces/Lie groups. The results will have a large impact because only a few types of such systems are presently known. Multi-component solitons: In work with students, we are studying interaction of 2-component mKdV solitons. Compared to the 1-component case, these solutions exhibit interesting, new features, including formation of "rogue waves", which is a very active topic in applied math. We will go on to n-component solitons. Conserved integrals: An open problem is to determine all conserved integrals for fundamental equations of fluid flow. In work with a PhD student on inviscid compressible fluid eqns., we derived all kinematic and vorticity integrals on moving domains in R^n. I recently extended the results to moving surfaces, yielding interesting generalizations of helicity and entropy circulation. I plan to obtain all such conserved integrals on moving domains/surfaces for several other important fluid eqns. in R^n and in Riemannian manifolds. The results will be of basic interest in the geometrical/topological study of n-dim. hydrodynamics elaborated by Arnold&Khesin. Symmetries, conservation laws, and exact solutions of nonlinear PDEs: Symmetry-invariant solutions are very useful for physical and mathematical reasons. With collaborators, I developed a novel symmetry method based on geometrical group foliations to find explicit invariant and non-invariant solutions to PDEs with power nonlinearities. For the nonlinear wave/heat equations we considered, these solutions have interesting analytical behavior related to blow-up, attractors, decay/dispersion, including cases of critical powers. I plan to continue developing this method and apply it to other physically and analytically interesting PDEs. This will have a large impact in symmetry methods and analysis of PDEs. I also plan to explore new connections between symmetries and conservation laws, stemming from work with Bluman summarized in our recent book.

近年来，偏微分方程组的对称性、守恒积分、哈密顿结构及其相关方面引起了人们的广泛关注。同时，对于可积偏微分方程组的研究也出现了大量的几何方法。我的建议在很大程度上建立在我目前拨款中的这些研究方向上。许多部分非常适合于硕士项目、博士论文和博士后工作，以及本科生的贡献。双哈密顿可积系统和曲线/曲面流：最近，我用几何框架方法导出了泛函多分量可积系统，它们是由黎曼对称空间和半单李群中的曲线流产生的具有双哈密顿结构的非线性薛定谔(NLS)、修正的Korteveg de Vries(MKdV)和Sine-Gordon(SG)方程的群不变推广。在与理科学生一起进行的工作中，我们正在派生新类型的NLS/mKdV方程。来自厄米特对称空间和洛伦兹对称空间。我计划从四元数和格拉斯曼空间中推导出新的四元数和矩阵类型的方程。我还计划将类似的方法应用于一个重要的公开问题，即如何从齐次空间中的曲线流导出群不变的可积系。我打算将我的方法扩展到导出Lax对。这将导致对Drinfeld&Sokolov关于仿射李代数中Lax对的普遍构造的几何理解，这是可积系统理论的核心结果。我还打算将Lax对和双哈密顿结构从框架变量提升回曲线流。这将给出Lax对和泊松括号上已知的抽象结果的显式表述。推广我在R^3中所做的工作，我计划从对称空间/李群中的几何曲面流导出2+1维可积系统。这一结果将产生很大的影响，因为目前已知的此类系统只有几种。多分量孤子：在与学生的合作中，我们正在研究两分量mKdV孤子的相互作用。与单分量情况相比，这些解决方案显示出有趣的新特征，包括形成“无赖波”，这是应用数学中一个非常活跃的话题。我们将继续讨论n分量孤子。守恒积分：一个悬而未决的问题是确定流体流动基本方程的所有守恒积分。在与一位研究无粘性可压缩流体方程的博士生合作中，我们导出了R^n中运动区域上的所有运动学积分和涡度积分。最近，我将结果推广到运动表面，得到了螺旋度和熵循环的有趣推广。我计划在其他几个重要的流体方程的移动区域/曲面上获得所有这样的守恒积分。在R^n和黎曼流形上。这些结果将对n维的几何/拓扑研究具有基本的意义。由Arnold&Khein阐述的流体力学。非线性偏微分方程解的对称性、守恒律和精确解：对称不变解在物理和数学上都是非常有用的。与合作者一起，我发展了一种新的基于几何群叶的对称方法来寻找具有幂非线性的偏微分方程组的显式不变解和非不变解。对于我们所考虑的非线性波动/热方程，这些解具有与爆破、吸引子、衰变/弥散有关的有趣的解析行为，包括临界功率的情况。我计划继续开发这种方法，并将其应用于其他物理和分析上感兴趣的PDE。这将对偏微分方程组的对称性方法和分析产生很大影响。我还计划探索对称性和守恒定律之间的新联系，这源于我们在最近的书中总结的与布鲁曼的工作。