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

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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.