Asymptotic stability of solitons of the gKdV equations with general nonlinearity

with general C nonlinearity f . Under an explicit condition on f and c > 0, there exists a solution in the energy space H of (0.1) of the type u(t, x) = Qc(x − x0 − ct), called soliton. In this paper, under general assumptions on f and Qc, we prove that the family of soliton solutions around Qc is asymptotically stable in some local sense in H , i.e. if u(t) is close to Qc (for all t ≥ 0), then u(t) locally converges in the energy space to some Qc+ as t → +∞. Note in particular that we do not assume the stability of Qc. This result is based on a rigidity property of equation (0.1) around Qc in the energy space whose proof relies on the introduction of a dual problem. These results extend the main results in [5], [6], [8] and [4], devoted to the pure power case.