Bright Solitary-Matter-Wave Collisions in a Harmonic Trap: Regimes of Soliton-like Behaviour

Systems of solitary-waves in the 1D Gross-Pitaevskii equation, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. To analyse the soliton-like nature of these solitary-waves, a particle analogy for the solitary-waves is formulated. Exact soliton solutions exist in the absence of an external trapping potential, which behave in a particle-like manner, and we find the particle analogy we employ to be a good model also when a harmonic trapping potential is present. In the case of two solitons, the particle model is integrable, and the dynamics are completely regular. The extension to three particles supports chaotic regimes. The agreement between the particle model and the wave dynamics remains good even in chaotic regimes. In the case of a system of two solitary waves of equal norm, the solitons are shown to retain their phase difference for repeated collisions. This implies that soliton-like regimes may be found in 3D geometries where solitary waves can be made to repeatedly collide out of phase, stabilising the condensate against collapse.