Asymptotic stability of solitons for subcritical generalized KdV equations

We prove in this paper the asymptotic completeness of the family of soliton s in the energy space for generalized Korteweg-de Vries equations in the su bcritical case (this includes in particular the KdV equation and the modifi ed KdV equation). This result is obtained as a consequence of a rigidity th eorem on the flow close to a soliton up to a scaling and a translation, whi ch has its own interest. The proofs use some tools introduced in a previous paper to prove similar results in the case of critical generalized KdV equ ation.