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

我的工作领域是微分方程的研究和计算机代数算法和程序的发展，以调查这些方程。偏微分方程（PDE）描述了一个量在空间和时间中的行为。如果它们允许传播并保持其形状的解决方案或具有形状保持结构的解决方案相互传播，相互穿透并随后再次采取其原始形状，则它们特别有趣。这样的偏微分方程有许多不寻常的性质，例如，它们有无穷多个守恒量和无穷多个无穷小对称性。它们被称为“可整合的”。为了找到这些可积的偏微分方程，人们以辅助方程的形式为它们的特殊性质制定数学条件，并试图求解它们。这些条件的共同点是它们是超定的，即它们涉及的条件比未知数多。这个想法是有一个强大的计算机程序包来解决超定系统，并反复使用它来找到不同类型的可积微分方程。这项工作在此授予申请有两个目的：进一步加强计算机代数软件包裂纹解决超定系统及其应用的可积性问题：*-反演递归和汉密尔顿算子 *-分类的可积演化型方程与费米子变量 *-可积常微分方程与矩阵变量 *-双汉密尔顿结构和泊松上同调的椭圆代数 * 一些可积偏微分方程确实有实际意义。人们可以制造玻璃纤维，使得光在这些纤维中的传播由可积分的薛定谔方程描述。这个方程的解（传播光脉冲）保持其形状，因此这种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.*****