Propagation, breathing and transition of matter-wave packet trains

We find a set of new exact solutions of a quantum harmonic oscillator, which describes some wave-packet trains with average energy being proportional to both the quantum level and classical energy of the oscillator. Center of the wave-packet trains may oscillate like a classical harmonic oscillator of frequency $\omega$. Width and highness of the trains may change simultaneously with frequency $2 \omega$ as an array of breathers. Under some perturbations the wave-packet trains could transit between the states of different quantum numbers. We demonstrate analytically and numerically that the wave-packet trains can be strictly fitted to the matter-wave soliton trains observed by Strecher et al. and reported in Nature 417, 150(2002). When the wave-packets breathe with greater amplitudes, they show periodic collapse and revival of the matter-wave.