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。

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.