Modified Scattering and Asymptotic Completeness for the Derivative Nonlinear Schrödinger equation

We prove a modified scattering and asymptotic completeness for the derivative nonlinear Schrödinger equation. This is the first result proving asymptotic completeness in a quasilinear setting. Our approach combines the method of testing by wave packets, introduced by Ifrim and Tataru, a bootstrap argument, and the Klainerman Sobolev vector field method.