Nonexistence of Asymptotically Free Solutions to Nonlinear Schrödinger Systems

We consider the nonlinear Schrödinger systems −i∂tu1 + 1 2 ∆u1 = F (u1, u2), i∂tu2 + 1 2 ∆u2 = F (u1, u2) in n space dimensions, where F is a p-th order local or nonlocal nonlinearity smooth up to order p, with 1 < p ≤ 1 + 2 n for n ≥ 2 and 1 < p ≤ 2 for n = 1. These systems are related to higher order nonlinear dispersive wave equations. We prove the non existence of asymptotically free solutions in the critical and sub-critical cases.