# American Institute of Mathematical Sciences

February  2010, 27(1): 1-24. doi: 10.3934/dcds.2010.27.1

## Vey theorem in infinite dimensions and its application to KdV

 1 CMLS, Ecole Polytechnique, 91128 Palaiseau, France, France

Received  September 2009 Revised  January 2010 Published  February 2010

We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,....$ 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 $j=1,2,....$ We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\rightarrow H$, 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 such that $F_j-$F j$=O(|u|^2)$ and each 1/2|F j|$^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism F$: H\rightarrow H$, the germ (F-id) is $\kappa$-smoothing, and each $I_j$ is an analytic function of the vector (1/2|Fj|$^2,j\ge1)$. 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'.
Citation: Sergei Kuksin, Galina Perelman. Vey theorem in infinite dimensions and its application to KdV. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 1-24. doi: 10.3934/dcds.2010.27.1
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