PAINLEVÉ ANALYSIS, LAX PAIR AND BÄCKLUND TRANSFORMATION FOR THE GROSS–PITAEVSKII EQUATION IN THE BOSE–EINSTEIN CONDENSATES

Due to their relevance to physics and technology, the Bose–Einstein condensates (BECs) are of current interest. Certain dynamics of the BECs, such as the cigar-shaped condensate confined in a cylindrically symmetric parabolic trap, can be described by the Gross–Pitaevskii (GP) equation with a time-dependent trap. In this paper, by virtue of the Painlevé analysis and symbolic computation, we derive the integrable condition for the GP equation with the time-dependent scattering length in the presence of a confining or expulsive time-dependent trap. Lax pair for this equation is directly obtained via the Ablowitz–Kaup–Newell–Segur scheme under the integrable condition. Bright one-soliton-like solution of the GP equation is presented via the Bäcklund transformation and some analytic solutions with variable amplitudes are obtained by the ansatz method. In addition, an infinite number of conservation laws are also derived. Those results could be of some value for the studies on the lower-dimensional condensates.