Perturbed KdV Equations
In this chapter we study small perturbations of the KdV equation $$ u_t = - u_{xxx} + 6uu_x $$ on the real line with periodic boundary conditions. We consider this equation as an infinite dimensional, integrable Hamiltonian system and subject it to suffic
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13 The Main Theorems In this chapter we study small perturbations of the KdV equation Ut
= -U xxx + 6uu x
on the real line with periodic boundary conditions. We consider this equation as an infinite dimensional, integrable Hamiltonian system and subject it to sufficiently small Hamiltonian perturbations. The aim is to show that large families of time-quasiperiodic solutions persist under such perturbations. This is true not only for this KdV equation, but in principle for all higher order KdV equations as well. As an example, the second equation in the KdV hierarchy will be considered in detail.
Background To set the stage we introduce for any integer N
~
0 the phase space
of real valued functions on SI = IR/Z, where
lIull~
= lu(O)1 2 + L Ikl2N lu(k)1 2 ke'Z.
is defined in terms of the Fourier transform u of u, u (x) = Lke'Z. U(k )e21rikx . In particular, we have Jeo = L2(SI) with norm 11·11 = 11·110' We endow JeN with the Poisson structure proposed by Gardner, {F,G} =
1aF SI
T. Kappeler et al., KdV & KAM © Springer-Verlag Berlin Heidelberg 2003
d aG
-----dx, ou(x) dx ou(x)
112
IV Perturbed KdV Equations
where F, G are differentiable functions on JfN with L 2-gradients in Jfl. The Hamiltonian corresponding to KdV is then given by
and
au at
=
d
dx
aH au
is the KdV equation written in Hamiltonian form. The Poisson bracket { . , . } is degenerate and admits the average [u] =
( u(x) dx
lSI
as a Casimir function. Moreover, the Poisson structure induces a trivial foliation with leaves c ER Instead of considering the restriction of the Hamiltonian H to each leaf Jf!', it is more convenient to choose a fixed phase space, Jf//, which is symplectomorphic to every other leaf Jf!' by translation. Writing u = v + c with [v] = 0 and c = [u] the Hamiltonian then takes the form
with
Hc(v) = { (!v;+v3)dx+6c ( !v 2 dx.
!
lSI
(13.1)
lSI
Here, lSI v2 dx is the zero-th Hamiltonian of the KdV hierarchy and corresponds to translation. To describe the structure of the phase space we recall some facts from the spectral theory of Hill's equation, with more details given in appendix B. For u in = Jfg consider the differential operator
L5
d2 L=--+u dx 2
on the interval [0, 2] with periodic boundary conditions. Its spectrum, denoted by spec(u), is pure point and consists of an unbounded sequence of periodic eigenvalues
Equality or inequality may occur in every place with a '::s' -sign, and one speaks of the gaps O"2n-1 (u), A,2n (u)) of the potential u and its gap lengths
n
~
1.
13 The Main Theorems
113
In the case of a double periodic eigenvalue, the gap is empty, and one speaks of a collapsed gap. Otherwise, the gap is said to be open. The space naturally decomposes into the isospectral sets
L5
Iso(u) = {v E
L5:
spec(v) = spec(u) }.
Our particular interest is in the isospectral sets of so called finite gap potentials u, which are characterized by the fact that only a finite number of gaps are open. Such see [49, 84]. potentials are known to be real analytic and dense in any space In particula
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