Integrability and tensor invariants in dynamics

动力学中的可积性和张量不变量

• 批准号: 138339-2010
• 负责人: Bogoyavlenskij, Oleg
• 金额: $ 1.46万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2010
• 资助国家: 加拿大
• 起止时间: 2010-01-01 至 2011-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=451809
• 关键词: Integrability; tensor; invariants; dynamics

The project is devoted to the theory and applications of tensor invariants of systems of partial differential equations (pde) and to the problems of integrability of pde's and dynamical systems. I will work on the Courant problems for systems of first order pde's. In papers [*2] and [*3], I have resolved the Courant problems on the decoupling and block-diagonalization of the systems with one spatial variable. I will apply the developed methods to study the Courant problems for systems with several spatial variables and the Courant problem on the invariant characterization of systems of conservation laws. Solution of these problems will use new invariants and identities derived in my papers [*5, 12] and will have applications for their numerical investigations. I will work on the extensions of Courant problems connected with existence of Hamiltonian and Bi-Hamiltonian structures for systems of hydrodynamic type. In collaboration with my PhD students, we will find invariant criteria that will be very useful for examination if a given system of pde's is Hamiltonian. Nowadays, such criteria are known only for the three-component systems due to our paper [7], jointly with P. Reynolds.

该项目致力于偏微分方程系统张量不变量的理论和应用，以及偏微分方程和动力系统的可积性问题。我将研究一阶偏微分方程系统的柯朗问题。在文[*2]和[*3]中，我解决了关于单空间变量系统的解耦和块对角化的Courant问题。我将应用开发的方法来研究的柯朗问题的系统与几个空间变量和柯朗问题的不变量表征系统的守恒律。这些问题的解决方案将使用新的不变量和身份在我的论文[* 5，12]中得出，并将其数值研究的应用。我将工作的扩展柯朗问题与存在的哈密顿和双哈密顿结构的系统的流体动力学类型。在与我的博士生合作，我们将找到不变的标准，这将是非常有用的检查，如果一个给定的系统偏微分方程是哈密尔顿。目前，由于我们与P.Reynolds联合发表的论文[7]，这种准则仅适用于三组分体系。