Preservation of the identities of multiple overlapping solitons through a collision with a strong integrability-breaking barrier

We show that when a strongly-coupled ``breather'' of the nonlinear Schr\"{o}dinger equation is scattered off of a nonperturbatively strong barrier, the solitons constituting the breather survive the collision. As the barrier height is lowered with impact speed held constant, at first all solitons are fully reflected; then suddenly the smallest soliton starts being fully transmitted; after the barrier is lowered some more, suddenly the next smallest soliton begins being fully transmitted as well, etc. Why the constituent solitons survive the integrability-breaking collision process is at present a mystery.