Investigation of the Hamiltonian Structure of the KdV System throughr-sMatrix Formalism Revealing Some New Aspects

Abstract

Poisson bracket (PB) structures of the KdV equation, which caused recent controversies are analysed in details through an alternative r and s matrix approach exploiting their nonultralocal canonical property. A symmetry criterion is found behind the existence of three different PB's. The required extensions of the PB are derived in a systematic way leading to the Faddeev-Takhtajan (FT) and the Arkadiev et al . brackets. This allows us to construct explicitly the action angle variables and compare their PB relations for all the brackets. It is revealed that for the FT bracket, contrary to the earlier result, the singularity free sector of the scattering matrix elements yields canonical action-angles, which also resolves the controversy regarding the nonseparability of the soliton modes from the continuous spectrum.