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We investigate the exact bright and dark solitary wave solutions of aneffective one dimensional (1D) Bose-Einstein condensate (BEC) by assuming thatthe interaction energy is much less than the kinetic energy in the transversedirection. In particular, following the earlier works in the literatureP\'erez-Garc\'ia et al. [Physica D 221 (2006) 31], Serkin et al. [Phys. Rev.Lett. 98 (2007) 074102], G\"urses [arXiv:0704.2435] and Kundu [Phys. Rev. E 79(2009) 015601], we point out that the effective 1D equation res ulting from theGross-Pitaevskii (GP) equation can be transformed into the stand ard soliton(bright/dark) possessing, completely integrable 1D nonlinear Schr\"o dinger(NLS) equation by effecting a change of variables of the coordinates and thewave function. We consider both confining and expulsive harmonic trappotentials separately and treat the atomic scattering length, gain/loss termand trap frequency as the experimental control parameters by modulating them asa function of time. In the case when the trap frequency is kept constant, weshow the existence of different kinds of soliton solutions, such as theperiodic oscillating solitons, collapse and revival of condensate, snake-likesolitons, stable solitons, soliton growth and decay and formation oftwo-soliton like bound state, as the atomic scattering length and gain/lossterm are varied. However when the trap frequency is also modulated, we show thephenomena of collapse and revival of two-soliton like bound state formation ofthe condensate for double modulated periodic potential and bright and darksolitons for step-wise modulated potentials.

Bright and dark solitons in a quasi 1D Bose-Einstein condensates modelled by 1D Gross-Pitaevskii equation with time-dependent parameters