GLOBAL WELL-POSEDNESS FOR THE NONLINEAR SCHRODINGER EQUATION WITH DERIVATIVE IN ENERGY SPACE

In this paper, we prove that there exists some small epsilon(*) > 0 such that the derivative nonlinear Schrodinger equation ( DNLS) is globally well-posed in the energy space, provided that the initial data u(0) is an element of H-1(R) satisfies parallel to u(0)parallel to(L2) < root 2 pi + epsilon(*). This result shows us that there are no blow-up solutions whose masses slightly exceed 2 pi, even if their energies are negative. This phenomenon is much different from the behavior of the nonlinear Schrodinger equation with critical nonlinearity. The technique used is a variational argument together with the momentum conservation law. Further, for the DNLS on the half-line R+, we show the blow-up for the solution with negative energy.