Abstract
This paper is concerned with the coupled nonlinear Schrodinger equations with harmonic potentials which describe the Bose-Einstein condensate under the magnetic trap. Two types of the invariant evolution flows are obtained, then a sharp criterion of global existence and blowing up of solutions for the equations is given in terms of the Hamiltonian invariants. Furthermore, we would answer: How small are the initial data such that the global solutions of the system exist?
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