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

Vey theorem in infinite dimensions and its application to KdV


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

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