In this note we give an overview of results concerning the Korteweg-de Vries equation ut=uxxx+6uux
and small perturbations of it. All the technical details are contained in our book [KdV & KAM, Springer, Berlin, 2003 MR1997070].
The KdV equation is an evolution equation in one space dimension which is named after the two Dutch mathematicians Korteweg and de Vries, but was apparently derived even earlier by Boussinesq. It was proposed as a model equation for long surface waves of water in a narrow and shallow channel. Their aim was to obtain as solutions solitary waves of the type discovered in nature by Scott Russell in 1834. Later it became clear that this equation also models waves in other homogeneous, weakly nonlinear and weakly dispersive media. Since the mid-sixties the KdV equation has received a lot of attention in the aftermath of the computational experiments of Kruskal and Zabusky, which led to the discovery of the interaction properties of the solitary wave solutions and in turn to the understanding of KdV as an infinite-dimensional integrable Hamiltonian system.
Our purpose here is to study small Hamiltonian perturbations of the KdV equation with periodic boundary conditions. In the unperturbed system all solutions are periodic, quasi-periodic, or almost periodic in time. The aim is to show that large families of periodic and quasi-periodic solutions persist under such perturbations. This is true not only for the KdV equation itself, but in principle for all equations in the KdV hierarchy. As an example, the second KdV equation is also considered.