On the Korteweg- de Vries Equation and KAM Theory

In this note we give an overview of results concerning the Korteweg-deVries equation ut = −uxxx + 6uux and small perturbations of it. All the technical details will be contained in our forth-coming book [27]. The KdV equation is an evolution equation in one space dimension which is named after the two Dutch mathematicians Korteweg and de Vries [29], but was apparently derived even earlier by Boussinesq [10, 57]. 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 Rus-sell [58] 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 received a lot of attention in the aftermath of the computational experiments of Kruskal and Zabusky [32], which lead to the discovery of the inter-action 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 equa-tion 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 will also be considered. 1 The KdV Equation Let us recall those features of the KdV equation that are essential for our purposes. It was observed by Gardner [21] and Faddeev & Zakharov [18] that the KdV equation can be written in the Hamiltonian form ∂u ∂