The limit equation for the Gross–Pitaevskii equations and S. Terraciniʼs conjecture

We establish the limit system for the Gross–Pitaevskii equations when the segregation phenomenon appears, and shows this limit is the one arising from the competing systems in population dynamics. This covers and verifies a conjecture of S. Terracini et al., both in the parabolic case and the elliptic case.