Soliton dynamics in an inhomogeneous Bose–Einstein condensate driven by linear potential

In this paper we study the propagation of solitons in a Bose–Einstein condensate governed by the time dependent one dimensional Gross–Pitaevskii equation managed by Feshbach resonance in a linear external potential. We give the Lax pair of the Gross–Pitaevskii equation in Bose–Einstein condensates and obtain exact N-soliton solution by employing the simple, straightforward Darboux transformation. As an example, we present exact one and two-soliton solution and discuss their transmission, interaction and dynamic properties. We further calculate the particle number, momentum and energy of the solitons and discuss their conservation laws. Knowledge of soliton dynamics helps us in understanding the physical nature of the condensate and in the calculation of the thermodynamic properties.