On the well-posedness problem for the derivative nonlinear Schrödinger equation

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