It is well known that the focusing nonlinear Schrödinger (fNLS) equation is an integrable PDE. When considered on the circle, the periodic eigenvalues of the Zakharov-Shabat (ZS) operator, appearing in the Lax pair formulation of the fNLS equation, form an infinite set of integrals of motion. In contrast to other integrable PDEs such as the defocusing nonlinear Schrodinger equation, the fNLS equation exhibits features of hyperbolic dynamics, in particular homoclinic orbits. In this talk I present a version of the Arnold-Liouville theorem for the fNLS equation: any connected level set of the above mentioned integrals of maximal - hence necessarily infinite dimension is a torus. On an invariant open neighborhood of such a torus we construct normal coordinates by developing the method of analytic continuation in the framework of normal form theory.

This is joint work with Peter Topalov.