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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569339
• 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. I will construct integrable Hamiltonian systems generalizing Volterra lattice and systems known in the literature as Bogoyavlenskij's lattices, using the method of descent recently developed in [6,10]. The novelty of the method is that it gives large families of Hamiltonian systems dynamics of which can be described in detail because they possess plenty of first integrals. I will work on the rigorous proof of integrability of these systems in the Liouville's sense and on the integrability of certain special cases in the exact form. I will study also integrability of several integro-differential equations obtained as continual limits of discrete integrable lattices. Jointly with my Master students, we will derive reductions of the ideal magnetohydrodynamics (MHD) equations with helical symmetry and will construct exact equipartition solutions [19] to the viscous MHD equations.

该项目致力于偏微分方程组（pde）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库朗问题。在论文 [*2] 和 [*3] 中，我解决了具有一个空间变量的系统解耦和块对角化的库朗问题。我将应用所开发的方法来研究具有多个空间变量的系统的库朗问题以及守恒定律系统的不变表征的库朗问题。这些问题的解决方案将使用我的论文 [*5, 12] 中导出的新的不变量和恒等式，并将应用于其数值研究。我将研究与流体动力类型系统的哈密顿和双哈密顿结构的存在相关的库朗问题的扩展。在与我的博士生合作时，我们将找到对于检查给定偏微分方程系统是否为哈密顿量的不变标准非常有用。如今，由于我们与 P. Reynolds 联合发表的论文 [7]，此类标准仅适用于三组分系统。 我将使用[6,10]中最近开发的下降方法构建可积哈密顿系统，推广 Volterra 晶格和文献中称为 Bogoyavlenskij 晶格的系统。该方法的新颖之处在于它给出了哈密顿系统动力学的大族，这些系统动力学可以被详细描述，因为它们拥有大量的一阶积分。我将致力于严格证明这些系统在刘维尔意义上的可积性，以及某些特殊情况的精确形式的可积性。我还将研究作为离散可积晶格的连续极限而获得的几个积分微分方程的可积性。与我的硕士生一起，我们将推导出具有螺旋对称性的理想磁流体动力学 (MHD) 方程的约简，并将构造粘性 MHD 方程的精确均分解 [19]。