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| We introduce some new results on Nonlinear Schr\"{o}dinger system arising from Bose-Einstein condensates. 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 verifies a conjecture of S. Terracini et.al., both in the parabolic case and the elliptic case.
We also study a class of nonlinear schr\"{o}dinger system with external sources terms, as these external sources terms are positive functions and small in some sense, we use Nehari manifold to get the existence of a positive ground state solution and a positive bound state solution. |
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