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

自然界中的过程，有时也包括社会中的过程，都是通过变量来描述的，比如“温度”、“电场”或“人口密度”。这些变量通过所谓的微分方程相互关联。如果通过这些方程描述的过程是复杂的，那么通常会发生的是一种混乱的情况，并不断增加，就像某人的家没有得到清理。但在极少数情况下，复杂的微分方程描述的过程是自动保持秩序的，即结构不会溶解或消散。这些结构被称为孤子。 例如，如果有人试图通过光纤电缆提交大量信息，并且不希望信息在途中混淆，那么所有这些都变得很重要。 在这项拨款建议的新类型的微分方程进行调查，有可能描述这样的孤子。寻找这些特殊的方程就像大海捞针。 强大的计算机程序已经开发出来，能够在方程中进行这种搜索。从特定搜索的角度来看，这种计算机程序的进一步改进是一种副作用。但由于这些计算机程序的普遍性，它们的进一步开发是其自身的一个子项目。 新发现的描述孤子的方程对进一步发展此类方程的理论具有重要意义。如果证明它们描述了物理过程，它们也将具有实际意义。沿着这条道路发展起来的计算机程序不仅可以处理数字，还可以处理整个公式和方程，适用于数学、自然科学和工程学。 拟议工作的一个更直接的目标是取得令人感兴趣的结果，在期刊上发表这些结果，并在国际会议上报告这些结果，从而在国际上展示加拿大的高水平研究。

Processes in nature and sometimes in society are described through variables, like 'temperature', 'electric field' or 'population density'. These variables are related to each other through so-called differential equations. If the process to be described through these equations is complicated then what typically happens is that a kind of disorder sets in and increases constantly, like someones home that does not get cleaned up from time to time. But in rare cases complicated differential equations describe processes where order is maintained automatically, i.e. where structures do not dissolve or dissipate. These structures are called solitons. All this becomes important, for example, if one tries to submit gigantic amounts of information through a fibre optics cable and does not want the information to get mixed up on its way. In this grant proposal new types of differential equations are to be investigated which have the potential to describe such solitons. Searching for these special equations is like looking for a needle in a haystack. Powerful computer programs have been developed that are able to perform such searches among equations. From the point of view of the specific search the further improvement of such computer programs is a side effect. But due to the universal nature of these computer programs their further development is a sub-project of its own. Newly found equations that describe solitons are of interest for the further development of the theory of such equations. They will also be of practical interest if it turns out that they describe physical processes. The computer programs that are developed along the way manipulate not only numbers but whole formulas and equations and are applicable across mathematics, the natural sciences and engineering. A more direct aim of the proposed work will be to obtain interesting results, to publish them in journals and report them on international conferences and thus to demonstrate internationally a high level of research in Canada.