We consider the derivative nonlinear Schrödinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and L 2 -critical with respect to scaling.The first question we discuss is whether ensembles of orbits with L 2equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction M (q) = |q| 2 < 4π. We conjecture that this restriction is unnecessary.Further, we prove that the problem is globally well-posed for initial data in H 1/6 under the same restriction on M . Moreover, we show that this restriction would be removed by a successful resolution of our equicontinuity conjecture.

No abstract

We prove that the derivative nonlinear Schrödinger equation in one space dimension is globally well-posed on the line in L 2 (R), which is the scaling-critical space for this equation.

No abstract

We study the defocusing inhomogeneous mass-critical nonlinear Schrödinger equation on R 2 iut + ∆u = g(nx)|u| 2 u for initial data in L 2 (R 2 ). We obtain sufficient conditions on g to ensure existence and uniqueness of global solutions for n sufficiently large, as well as homogenization.