DYNAMICS OF STRONGLY INTERACTING UNSTABLE TWO-SOLITONS FOR GENERALIZED KORTEWEG-DE VRIES EQUATIONS

We consider the generalized Korteweg-de Vries equation ∂tu = −∂x(∂_x²u + f(u)), where f is an odd function of class C³. Under some assumptions on f, this equation admits solitary waves, that is solutions of the form u(t, x) = Qv(x− vt− x0), for v in some range (0, v∗). We study pure two-solitons in the case of the same limit speed, in other words global solutions u(t) such that (∗) _tlim _→∞ ∥u(t)−(Qv(· −x1(t))±Qv(· −x2(t)))∥_H1 = 0, _tlim _→∞ x2(t)−x1(t) = ∞. Existence of such solutions is known for f(u) = |u|^p−1u with p ∈ Z \ {5} and p > 2. We describe the dynamical behavior of any solution satisfying (∗) under the assumption that Qv is linearly unstable (which corresponds to p > 5 for power nonlinearities). We prove that in this case the sign in (∗) is necessarily “+”, which corresponds to an attractive interaction. We also prove that the distance x2(t) − x1(t) between the solitons equals (Formula presented) for some κ = κ(v) > 0.