Asymptotic stability of solitons of the gKdV equations with general nonlinearity

AbstractWe consider the generalized Korteweg-de Vries equation (gKdV) $$\partial_t u + \partial_x (\partial_x^2 u + f(u)) = 0, \quad (t, x) \in [0, T) \times {\mathbb{R}},$$with general C3 nonlinearity f. Under an explicit condition on f and c > 0, there exists a solution in the energy space H1 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 solitons around Qc is asymptotically stable in some local sense in H1, 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 the gKdV equation around Qc 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.