Two soliton collision for nonlinear Schrödinger equations in dimension 1

We study the collision of two solitons for the nonlinear Schrödinger equation iψt=−ψxx+F(2|ψ|)ψ, F(ξ)=−2ξ+O(ξ2) as ξ→0, in the case where one soliton is small with respect to the other. We show that in general, the two soliton structure is not preserved after the collision: while the large soliton survives, the small one splits into two outgoing waves that for sufficiently long times can be controlled by the cubic NLS: iψt=−ψxx−22|ψ|ψ.