Applications of advances in computer algebra to studying classical integrable systems and related algebraic structures

我的工作领域是研究微分方程，以及开发计算机代数算法和程序来研究这些方程。偏微分方程(PDE)描述一个量在空间和时间上的行为。如果它们允许传播并保持其形状的解决方案或具有保形结构的解决方案彼此传播，相互穿透，然后重新恢复原始形状，则它们尤其有趣。这类偏微分方程具许多不寻常的性质，例如，它们有无穷多个守恒量和无穷多个无穷小对称。它们被称为“可积的”。为了找到这些可积的偏微分方程组，我们用辅助方程的形式为它们的特殊性质建立了数学条件，并试图求解它们。这些条件的共同之处在于它们都是过度确定的，即它们涉及的条件比未知条件更多。这个想法是要有一个强大的计算机程序包来求解超定系统，并反复使用它来寻找不同类型的可积微分方程。 这项赠款申请的工作有两个目标：进一步加强用于解决超定系统的计算机代数软件包CRECH，并将其应用于可积性问题： 递归算子和哈密顿算子的逆 -含费米子变量的可积发展型方程的分类 -带矩阵变量的可积常数 椭圆代数的双哈密顿结构和Poisson上同调 一些可积的偏微分方程组确实具有实际意义。人们可以制造玻璃纤维，使得光在这些光纤中的传播可以用可积薛定谔方程来描述。这个方程有解决方案(传播的光脉冲)保持它们的形状，因此这样的电缆非常适合长距离传输大量数据。

The area of my work is the study of differential equations and the development of computer algebra algorithms and programs to investigate these equations. Partial differential equations (PDEs) describe the behaviour of a quantity in space and time. They are especially interesting if they allow solutions that propagate and keep their shape or solutions that have shape preserving structures propagating towards each other, penetrating each other and afterwards taking again their original shape. Such PDEs have a number of unusual properties, for example, they have infinitely many conserved quantities and infinitely many infinitesimal symmetries. They are called 'integrable'. To find these integrable PDEs one formulates mathematical conditions for their special properties in the form of auxiliary equations and tries to solve them. These conditions have in common that they are overdetermined, i.e. they involve more conditions than unknowns. The idea is to have one powerful package of computer programs to solve overdetermined systems and to use that repeatedly to find integrable differential equations of different kind. Work under this grant application has two aims: the further strengthening of the computer algebra package Crack for solving overdetermined systems and its application to integrability problems: - Inversion of Recursion and Hamiltonian Operators - Classification of integrable evolution-type equations with fermionic variables - Integrable ODEs with matrix variables - Bi-hamiltonian structures and Poisson cohomologies of Elliptic Algebras Some integrable PDEs do have a practical importance. One can manufacture glass fibers such that light propagation in these fibers is described by the integrable Schroedinger equation. This equation has solutions (propagating pulses of light) that keep their shape, and thus such cables are perfectly suited to transmitting large amounts of data over long distances.