Poisson structure for the KdV equation (Q1077631)

The authors show that the usual Poisson structure for the KdV equation is degenerate. This kind of Poisson structure was introduced first by Gardner and modified in this paper to include the case when the variational derivatives of functionals have nonzero limiting values. Basing on the above observation the authors make a remark that the usual interpretation of some quantities, coming from the scattering data, as action-angle variables should be revised. The corrected Poisson brackets of these quantities are given. The reviewer would like to mention that the degeneracy of the symplectic structure of the KdV equation is also discussed in a paper by \textit{Y. Nutku} [J. Math. Phys. 25, 2007-2008 (1984)].