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On the Korteweg-de Vries equation: Convergent Birkhoff normal form. (English) Zbl 0868.35099

The authors consider the Korteweg-de Vries (KdV) equation

${V}_{t}+{V}_{xxx}-6V·{V}_{x}=0$

on the circle, i.e., for functions $V=V\left(x,t\right)$ on ${S}^{1}×ℝ$. It is well-known that the KdV equation represents a completely integrable Hamiltonian system (of infinite dimension) with phase space ${H}^{n}\left({S}^{1};ℝ\right)$, where $n$ is a positive integer, and with a Poisson structure inducing a symplectic structure on the phase space.

The main result of the present paper states that there is a symplectic morphism from the zero-leaf of the phase space, with respect to the Casimir function defined by the Poisson structure to some symplectic Hilbert space. This symplectomorphism is bijective, real-analytic in both directions, and induces globally defined real-analytic action-angle coordinates for the entire KdV hierarchy. This theorem provides the (apparently) first infinite-dimensional counterpart of the result by J. Vey [Am. J. Math. 100, 591-614 (1978; Zbl 0384.58012)], which states that a finite-dimensional completely integrable Hamiltonian system with real-analytic Hamiltonian functional admits action-angle variables in a neighborhood of a non-resonant elliptic fixed point.

In the second part of the paper, the authors apply their main result to the study of the regularity of the KdV-Hamiltonian vector field. In particular, the local existence of a convergent Birkhoff normal form is proved.

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems