ON THE SYMPLETIC STRUCTURE OF THE PHASE-S PACE FOR THE PERIODIC KORTEWEG-DEVRIES EQUATION

We prove that the generalized phase space of KdV on S-1, i.e. (L(0)(2) ([0, 1]), omega(G)) where omega(G) denotes the Gardner symplectic stru cture on the space L(0)(2)([0, 1]) of L(2) functions with average 0, i s symplectomorphic to the phase space (l(1/2)(2)(R(2)),omega(0) of inf initely many harmonic oscillators, where l(1/2)(2)(R(2)) denotes the H ilbert space of sequences (x(n),y(n))(n) greater than or equal to 1 sa tisfying Sigma n(x(n)(2)+ y(n)(2))<infinity, endowed with the canonica l symplectic structure omega(0). The symplectomorphism Omega(n greater than or equal to 1) from (L(0)(2)([0, 1], omega(G)) onto (l(1/2)(2)(R (2)), omega(0)) is shown to be bianalytic. Similar results hold for th e periodic Toda equations and the periodic nonlinear Schrodinger equat ion (defocusing).