Vey theorem in infinite dimensions and its application to KdV

TL;DR

This paper proves a Vey-type theorem for infinite-dimensional Hamiltonian systems, showing the existence of a Birkhoff normal form for KdV with a 1-smoothing symplectomorphism near the origin.

Contribution

It extends Vey's theorem to infinite dimensions, demonstrating the existence of a smoothing symplectomorphism that simplifies the KdV equation near equilibrium.

Findings

Existence of a modified integrable structure with smoothing properties.

Application of the theorem to establish Birkhoff normal form for KdV.

The integrating transformation is a 1-smoothing analytic map.

Abstract

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 $I_j={1/2}|F_j|^2$, where $F_j:H\to \R^2$, $F_j(0)=0$ for $j=1,2,...$ . We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\to H$, 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 $F^\prime: H\to H$, 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'.