In this paper we provide new asymptotic estimates of various spectral quantities of Zakharov-Shabat operators on the circle. These estimates are uniform on bounded subsets of potentials in Sobolev spaces.

In this paper we prove approximation properties of the solutions of the defoucsing NLS equation on the circle by nearly linear flows. In addition we show that spatially periodic solutions of the defocusing NLS equation evolving in fractional Sobolev spaces H^s with s≥ 1 remain bounded for all times.

In this paper we prove new qualitative features of solutions of KdV on the circle. The first result says that the Fourier coefficients of a solution of KdV in Sobolev space H N , N ≥ 0, admit a WKB type expansion up to first order with strongly oscillating phase factors defined in terms of the KdV frequencies. The second result provides estimates for the approximation of such a solution by trigonometric polynomials of sufficiently large degree.

In this paper we provide new asymptotic estimates of the Floquet exponents of Schrodinger operators on the circle. By the same techniques, known asymptotic estimates of various others spectral quantities are improved.

We show that the modified Korteweg-de Vries equation with periodic boundary conditions admits global Birkhoff coordinates on any Soblev space HN with N≥1. The construction of these coordinates involves a new Lax pair for mKdV.