Nonautonomous solitons in terms of the double Wronskian determinant for a variable-coefficient Gross-Pitaevskii equation in the Bose-Einstein condensate

摘要

Under investigation in this paper is a variable-coefficient Gross-Pitaevskii equation which describes the Bose-Einstein condensate. Lax pair, bilinear forms and bilinear Backlund transformation for the equation under some integrable conditions are derived. Based on the Lax pair and bilinear forms, double Wronskian solutions are constructed and verified. The Nth-order nonautonomous solitons in terms of the double Wronskian determinant are given. Propagation and interaction for the first- and second-order nonautonomous solitons are discussed from three cases. Amplitudes of the first- and second-order nonautonomous solitons are affected by a real parameter related to the variable coefficients, but independent of the gain-or-loss coefficient m(0)(t) and linear external potential coefficient m(1)(t). For Case 1 left perpendicularm(0)(t) = 0right perpendicular, m(1)(t) leads to the accelerated propagation of nonautonomous solitons. Parabolic-, cubic-, exponential-and cosine-type nonautonomous solitons are exhibited due to the different choices of m(1)(t). For Case 2 [m(1)(t) = 0], if the real part of the spectral parameter equals 0, stationary soliton can be formed. If we take the harmonic external potential coefficient m(2)(t) as a positive constant and let the real parts of the two spectral parameters be the same, bound-state-like structures can be formed, but there are only one attractive and two repulsive procedures. For Case 3 [m(0)(t) and m(1)(t) are taken as nonzero constants], head-on interaction, overtaking interaction and bound-state structure can be formed based on the signs of the two spectral parameters.