On the Korteweg-de Vries equation: Convergent Birkhoff normal form
The authors consider the Korteweg-de Vries (KdV) equation \[ V_t+ V_{xxx}- 6V\cdot V_x=0 \] on the circle, i.e., for functions \(V=V(x,t)\) on \(S^1\times\mathbb{R}\). It is well-known that the KdV equation represents a completely integrable Hamiltonian system (of infinite dimension) with phase space \(H^n(S^1;\mathbb{R})\), 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 \textit{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.
scientific article; zbMATH DE number 934260