Integrability and Multi-Soliton Solutions for a Variable-Coefficient Coupled Gross--Pitaevskii System for Atomic Matter Waves

Under investigation in this paper is a variable-coefficient coupled Gross–Pitaevskii (GP) system, which is associated with the studies on atomic matter waves. Through the Painlevé analysis, we obtain the constraint on the variable coefficients, under which the system is integrable. The bilinear form and multi-soliton solutions are derived with the Hirota bilinear method and symbolic computation. We found that: (i) in the elastic collisions, an external potential can change the propagation of the soliton, and thus the density of the matter wave in the two-species Bose–Einstein condensate (BEC); (ii) in the shape-changing collision, the solitons can exchange energy among different species, leading to the change of soliton amplitudes. We also present the collisions among three solitons of atomic matter waves.