  We study generalized variants of the Burgers equation and the KdV equation on
the circle. The main goal of the paper is to show that both extensions can be
recast as geodesic equations on a suitable diffeomorphism group of the circle
and the corresponding Bott-Virasoro group respectively. As a consequence we
obtain that the initial value problem for the Burgers equation with an
additional quadratic term is well-posed on a scale of Sobolev spaces on the
circle.
