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.