Abstract
This paper is concerned with the supercritical nonlinear Schrödinger equation with a harmonic potential which describes the attractive Bose-Einstein condensate under a magnetic trap. We establish two types of new invariant evolution flows. Then, in terms of the Hamiltonian invariants, we derive a new sharp energy criterion for global existence and blowing up of solutions of the equation, which can be precisely computed in the Bose-Einstein condensate.
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