Asymptotic stability of solitons of the gKdV equations with general nonlinearity

We consider the generalized Korteweg-de Vries equation (gKdV) with general C ³ nonlinearity f. Under an explicit condition on f and c > 0, there exists a solution in the energy space H ¹ of the type u(t, x) = Q _c (x − x ₀ − ct), called soliton. In this paper, under general assumptions on f and Q _c , we prove that the family of solitons around Q _c is asymptotically stable in some local sense in H ¹, i.e. if u(t) is close to Q _c (for all t ≥  0), then u(t) locally converges in the energy space to some Q _c+ as t → +∞. Note in particular that we do not assume the stability of Q _c . This result is based on a rigidity property of the gKdV equation around Q _c in the energy space whose proof relies on the introduction of a dual problem. These results extend the main results in Martel (SIAM J. Math. Anal. 38:759–781, 2006); Martel and Merle (J. Math. Pures Appl. 79:339–425, 2000), (Arch. Ration. Mech. Anal. 157:219–254, 2001), (Nonlinearity 1:55–80), devoted to the pure power case. This research was supported in part by the Agence Nationale de la Recherche (ANR ONDENONLIN).