Exact solitonic solutions of the one dimensional gross-pitaevskii equation with a time-dependent harmonic potential and interatomic interaction

We derive exact solitonic solutions of the one-dimensional time-dependent Gross-Pitaevskii equation with time-dependent strengths of the harmonic external potential and the interatomic interaction. The time-dependence of the external potential and interatomic interaction are given in terms of a general function of time. For an oscillating strength of the external potential, the solutions correspond to breathing single and multiple solutions. The amplitude and frequency of the oscillating potential can be used to control the dynamics of the centre of mass of the solitons. For certain values of these parameters, the solitons can be trapped at the centre of the harmonic potential.