PERIODIC NONLINEAR SCHRODINGER-EQUATION AND INVARIANT-MEASURES

In this paper we continue some investigations on the periodic NLSE iu(t) + u(xx) + u\u\(p-2) = 0 (p less-than-or-equal-to 6) started in [LRS]. We prove that the equation is globally wellposed for a set of data phi of full normalized Gibbs measure e(-betaH(phi)) Hd phi(x), H (phi) = 1/2 integral \phi'\2 - 1/p (after suitable L2-truncation). The set and the measure are invariant under the flow. The proof of a similar result for the KdV and modified KdV equations is outlined. The main ingredients used are some estimates from [B1] on periodic NLS and KdV type equations.