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In this paper, we consider the long time dynamics of radially symmetricsolutions of nonlinear Schr\"odinger equations (NLS) having a minimal massground state. In particular, we show that there exist solutions with initialdata near the minimal mass ground state that oscillate for long time. Moreprecisely, we introduce a coordinate defined near the minimal mass ground statewhich consists of finite and infinite dimensional part associated to thediscrete and continuous part of the linearized operator. Then, we show that thefinite dimensional part, two dimensional, approximately obeys Newton's equationof motion for a particle in an anharmonic potential well. Showing that theinfinite dimensional part is well separated from the finite dimensional part,we will have long time oscillation.

Long time oscillation of solutions of nonlinear Schr\"odinger equations near minimal mass ground state