  We study the class of generalized Korteweg-DeVries equations derivable from
the Lagrangian: $ L(l,p) = \int \left( \frac{1}{2} \vp_{x} \vp_{t} - {
{(\vp_{x})^{l}} \over {l(l-1)}} + \alpha(\vp_{x})^{p} (\vp_{xx})^{2} \right)
dx, $ where the usual fields $u(x,t)$ of the generalized KdV equation are
defined by $u(x,t) = \vp_{x}(x,t)$. This class contains compactons, which are
solitary waves with compact support, and when $l=p+2$, these solutions have the
feature that their width is independent of the amplitude. We consider the
Hamiltonian structure and integrability properties of this class of KdV
equations. We show that many of the properties of the solitary waves and
compactons are easily obtained using a variational method based on the
principle of least action. Using a class of trial variational functions of the
form $u(x,t) = A(t) \exp \left[-\beta (t) \left|x-q(t) \right|^{2n} \right]$ we
find soliton-like solutions for all $n$, moving with fixed shape and constant
velocity, $c$. We show that the velocity, mass, and energy of the variational
travelling wave solutions are related by $ c = 2 r E M^{-1}$, where $ r =
(p+l+2)/(p+6-l)$, independent of $n$.\newline \newline PACS numbers: 03.40.Kf,
47.20.Ky, Nb, 52.35.Sb
