Abstract

We study the problem of two-soliton collision for the generalized Korteweg-de Vries equations, completing some recent works of Y. Martel and F. Merle [22, 25]. 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 collision are precisely the integrable cases, namely, the quadratic (KdV), cubic (mKdV), and Gardner nonlinearities.