We consider the generalized Korteweg-de Vries equation ∂tu + ∂ 3 xu + ∂x(u ) = 0, (t, x) ∈ R2, in the supercritical case p > 5, and we are interested in solutions which converge to a soliton in large time in H. In the subcritical case (p < 5), such solutions are forced to be exactly solitons by variational characterization [1, 19], but no such result exists in the supercritical case. In this pa...

We study the problem of 2-soliton collision for the generalized Korteweg-de Vries equations, completing some recent works of Y. Martel and F. Merle [22, 23]. We classify the nonlinearities for which collisions are elastic or inelastic. Our main result states that in the case of small solitons, with one soliton smaller than the other one, the unique nonlinearities allowing a perfectly elastic co...

We prove in this paper the stability and asymptotic stability in H of a decoupled sum of N solitons for the subcritical generalized KdV equations ut + (uxx + u )x = 0 (1 < p < 5). The proof of the stability result is based on energy arguments and monotonicity of local L norm. Note that the result is new even for p = 2 (the KdV equation). The asymptotic stability result then follows directly fro...

We consider the (KdV)/(KP-I) asymptotic regime for the Nonlinear Schrödinger Equation with a general nonlinearity. In a previous work, we have proved the convergence to the Korteweg-de Vries equation (in dimension 1) and to the Kadomtsev-Petviashvili equation (in higher dimensions) by a compactness argument. We propose a weakly transverse Boussinesq type system formally equivalent to the (KdV)/...

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‖...