  This is the extended version of a lecture course given at the University of
Vienna in the spring term 2005. The main aim of this course was to understand
the papers \cite{10} and \cite{11} and to give a complete account of existence
and uniqueness of the solutions of the members of higher order of the
hierarchies of Burgers' equation and the Korteweg-de Vries equation, including
their derivation and all the necessary background, both on the circle and on
the real line in the setting of rapidly decreasing functions. These are all
geodesic equations of infinite dimensional regular Lie groups, namely the
diffeomorphism group of the line or the circle and the corresponding Virasoro
group.
