Refined asymptotics around solitons for gKdV equations

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. Stability theory for Qc is well-known. In [11], [14], we have proved that for f(u) = u, p = 2, 3, 4, the family of solitons 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, . + ρ(t)) locally converges in the energy space to some Qc+ as t → +∞, for some c ∼ c. The main improvement in [14] is a direct proof, based on a localized Viriel identity on the solution u(t). As a consequence, we have obtained an integral estimate on u(t, .+ ρ(t)) −Qc+ as t → +∞. In [9] and [15], using the indirect approach of [11], we could extend the asymptotic stability result under general assumptions on f and Qc. However, without Viriel argument directly on the solution u(t), no integral estimate is available in that case. The objective of this paper is twofold. The main objective is to prove that in the case f(u) = u, p = 2, 3, 4, ρ(t) − c+t has limit as t → +∞ under the additional assumption x+u ∈ L(R), which is consistent with a counterexample in [14]. This result persists for general nonlinearity if a Virial type estimate is assumed. The main motivation for this type of result is the determination of explicit shifts due to collision of two solitons in the nonintegrable case p = 4, see [16]. The second objective of this paper is to provide large time stability and asymptotic stability results for two soliton solutions for the case of general nonlinearity f(u), when the ratio of the speeds of the solitons is small. The motivation is to accompany the two papers [16], [17], devoted to collisions of two solitons in the nonintegrable case. The arguments are refinements of [22], [18] specialized to the case u(t) ∼ Qc1 +Qc2 , for 0 < c2 ≪ c1.