The Beginning
In this book we consider the Korteweg-de Vries (KdV) equation $${u_t} = - {u_{xxx}} + 6u{u_x}.$$.
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The Beginning
1 Overview In this book we consider the Korteweg-de Vries (KdV) equation Ut
=
-U xxx
+ 6uu x ·
The KdV equation is an evolution equation in one space dimension which is named after the two Dutch mathematicians Korteweg and de Vries [66] - see also Boussinesq [18] and Rayleigh [113]. 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 Russell [114] 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 [69], which lead to the discovery of the interaction properties of the solitary wave solutions and in tum 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 will also be considered.
The KdV Equation Let us recall those features of the KdV equation that are essential for our purposes. It was observed by Gardner [46], see also Faddeev & Zakharov [40], that the KdV equation can be written in the Hamiltonian form
au at T. Kappeler et al., KdV & KAM © Springer-Verlag Berlin Heidelberg 2003
=
d dx
aH
au
2
I The Beginning
with the Hamiltonian
H(u) = { (!u;
lSI
+ U 3 ) dx,
where oH lou denotes the L 2-gradient of H, representing the Frechet derivative of H with respect to the standard scalar product on L 2. Since we are interested in spatially periodic solutions, we take as the underlying phase space the Sobolev space
of real valued functions with period 1, where N the Poisson bracket proposed by Gardner,
{F,G}= (
lSI
~
1 is an integer, and endow it with
~~~dx. ou(x) dx ou(x)
Here, F and G are differentiable functions on JeN with L 2-gradients in Je l . This makes JeN a Poisson manifold, on which the KdV equation may also be represented in the form Ut = {u, H} familiar from classical mechanics. We note that the initial value problem for the KdV equation on the circle S I is well posed on every Sobolev space JeN with N ~ 1: for initial data UO E JeN it has been shown by Temam for N = 1, 2 [128] and by Saut & Temam for any real N ~ 2 [121] that there exists a unique solution evolving in JeN and defined globally in time. For further results on the initial value problem see for instance [78, 88, 126] as well as the more recent results [14, 15,64]. The KdV equation admits infinitely many conserved quantities, or integrals, in involution, an
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