Two soliton collision for nonlinear Schrödinger equations in dimension 1

Abstract

We study the collision of two solitons for the nonlinear Schrödinger equation $i{\psi }_t=-{\psi }_{xx}+F(|\psi {|}^2)\psi$, $F(\xi )=-2\xi +O({\xi }^2)$ as $\xi \to 0$, 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: $i{\psi }_t=-{\psi }_{xx}-2|\psi {|}^2\psi$.