ar X iv : 0 70 9 . 26 77 v 1 [ m at h . A P ] 1 7 Se p 20 07 Stability of two soliton collision for nonintegrable gKdV equations ∗

We continue our study of the collision of two solitons for the subcritical generalized KdV equations ∂tu+ ∂x(∂ 2 xu+ f(u)) = 0. (0.1) Solitons are solutions o the type u(t, x) = Qc0(x− x0 − c0t) where c0 > 0. In [21], mainly devoted to the case f(u) = u, we have introduced a new framework to understand the collision of two solitons Qc1 , Qc2 for (0.1) in the case c2 ≪ c1 (or equivalently, ‖Qc2‖H1 ≪ ‖Qc1‖H1). In this paper, we consider the case of a general nonlinearity f(u) for which Qc1 , Qc2 are nonlinearly stable. In particular, since f is general and c1 can be large, the results are not pertubations of the ones for the power case in [21]. First, we prove that the two solitons survive the collision up to a shift in their trajectory and up to a small perturbation term whose size is explicitely controlled from above: after the collision, u(t) ∼ Q c + 1 +Q c + 2 where c+j is close to cj (j = 1, 2). Then, we exhibit new exceptional solutions similar to multi-soliton solutions: for all c1, c2 > 0, c2 ≪ c1, there exists a solution φ(t) such that φ(t, x) = Qc1(x−ρ1(t)) +Qc2(x−ρ2(t)) + η(t, x), for t ≪ −1, φ(t, x) = Qc1(x−ρ1(t)) +Qc2(x−ρ2(t)) + η(t, x), for t ≫ 1, where ρj(t) → cj (j = 1, 2) and η(t) converges to 0 in a neighborhood of the solitons as t → ±∞. The analysis is splitted in two distinct parts. For the interaction region, we extend the algebraic tools developed in [21] for the power case, by expanding f(u) as a sum of powers plus a perturbation term. To study the solutions in large time, we rely on previous tools on asymptotic stability in [17], [22] and [18], refined in [19], [20]. ∗This research was supported in part by the Agence Nationale de la Recherche (ANR ONDENONLIN).