This paper develops the Hirota method carefully for applying into the growing model of quasi-one-dimensional Bose Einstein condensations with attractive and repulsive interaction, respectively. After a tedious calculation it obtains the exact bright and dark soliton solutions analytically. It shows that the growing model has the important effect on the soliton amplitude and the time-dependent potential only contributes to the phase and phase velocity. A detailed analysis for the asymptotic behaviour of two-soliton solutions shows that the collision of two soliton is elastic.
We investigate the combined soliton solutions of two-component Bose-Einstein condensates with external potential. The ＂phase diagram＂ is obtained for the formation regions of different combined solitons. Our results show that the intraspecies （interspecies） interaction strengths and the external trapped potential clearly affect the formation of dark- dark, bright-bright, and dark bright soliton solutions in different regions. Especially, we find that the bright bright （dark^dark） soliton can exist in the case of both repulsive （attractive） intraspecies interaction strengths in the presence of external potential. This novel phenomenon is completely different from the formation of soliton solution of one- component Bose-Einstein condensates without external potential, and it will be useful for the study of two-component Bose-Einstein condensates.
