Sergei Kuksin and Galina Perelman
Vey theorem in infinite dimensions and its application to KdV
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ABSTRACT. We consider an integrable infinite-dimensional
Hamiltonian system in a Hilbert space
$H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;.\dots)\}$
with integrals $I_1, I_2,\dots$ which can be written
as $I_j=\frac{1}{2}|F_j|^2$, where $F_j:H\rightarrow \R^2$,
$F_j(0)=0$ for $j=1,2,\dots$\,.
We assume that the maps $F_j$ define a germ of an analytic diffeomorphism
$F=(F_1,F_2,\dots):H\rightarrow H$,
such that $dF(0)=id$, $(F-id)$ is a $\kappa$-smoothing map ($\kappa\geq 0$)
and some other mild restrictions on $F$ hold.
Under these assumptions we show that the maps $F_j$
may be modified to maps $F_j^\prime$ such that
$F_j-F_j^\prime=O(|u|^2)$ and each $\frac12|F'_j|^2$ still
is an integral of motion.
Moreover, these maps jointly
define a germ of an analytic
symplectomorphism $F^\prime: H\rightarrow H$,
the germ $(F^\prime-id)$ is $\kappa$-smoothing, and each $I_j$ is
an analytic function of the vector $(\frac12|F'_j|^2,j\ge1)$.
Next we show that the theorem with $\kappa=1$
applies to the KdV equation.
It implies that in the vicinity of the origin in a functional space
KdV admits the Birkhoff normal form and the integrating
transformation has the form
`identity plus a 1-smoothing analytic map'.