An Additional Gibbs' State for the Cubic Schrodinger Equation on the   Circle
  An invariant Gibbs' state for the nonlinear Schrodinger equation on the circle was constructed by Bourgain, and McKean, out of the basic Hamiltonian using a trigonometric cut-off. The cubic nonlinear Schrodinger equation is a completely integrable system having an infinite number of additional integrals of motion. In this paper we construct the second invariant Gibbs' state from one of these additional integrals for the cubic NLS on the circle. This additional Gibbs' state is singular with respect to the Gibbs' state previously constructed from the basic Hamiltonian. Our approach employs the Ablowitz-Ladik system, a completely integrable discretization of the cubic Schrodinger equation. 
K.L. Vaninsky