Symplectic structures and volume elements in the function space for the cubic Schrödinger equation

Abstract

We consider various trace formulas for the cubic Schrodinger equa- tion in the space of infinitely smooth functions subject to periodic boundary conditions. The formulas relate conventional integrals of motion to the periods of some Abelian differentials (holomorphic one-forms) on the spectral curve. We show that the periods of Abelian differentials are global coordinates on the moduli space of spectral curves. The exterior derivatives of the holomorphic one-forms are the basic and higher symplectic structures on the phase space. We write ex- plicitly these symplectic structures in QP coordinates. We compute the ratio of two symplectic volume elements in the infinite genus limit.