Integrability and tensor invariants in dynamics

本课题主要研究偏微分方程组张量不变量的理论与应用，以及偏微分方程组与动力系统的可积性问题。我将研究一阶微分方程系统的柯朗问题。在论文[*2]和[*3]中，我解决了单空间变量系统解耦和块对角化的Courant问题。我将运用已发展的方法研究具有多个空间变量的系统的Courant问题和关于守恒律系统的不变表征的Courant问题。这些问题的解决将使用我在论文[* 5,12]中导出的新的不变量和恒等式，并将应用于它们的数值研究。我将研究与水动力型系统的哈密顿和双哈密顿结构存在性有关的Courant问题的扩展。在和我的博士生的合作中，我们会找到一些不变的准则，这些准则对于检验一个给定的偏微分方程是否为哈密顿函数非常有用。如今，由于我们与P. Reynolds合著的论文[7]，这些准则仅在三组分系统中已知。

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.