  We study the Derivative Nonlinear Schr\"odinger equation for general initial
conditions in weighted Sobolev spaces that can support bright solitons (but
excluding spectral singularities). We prove global well-posedness and give a
full description of the long- time behavior of the solutions in the form of a
finite sum of localized solitons and a dispersive component. At leading order
and in space-time cones, the solution has the form of a multi-soliton whose
parameters are slightly modified from their initial values by soliton-soliton
and soliton-radiation interactions. Our analysis provides an explicit
expression for the correction dispersive term. We use the nonlinear steepest
descent method of Deift and Zhou revisited by the $\bar{\partial}$-analysis of
Dieng-McLaughlin and complemented by the recent work of
Borghese-Jenkins-McLaughlin on soliton resolution for the focusing nonlinear
Schr\"odinger equation.
