Darboux transformation of nonisospectral coupled Gross-Pitaevskii equation and its multi-component generalization

We extend one component Gross-Pitaevskii equation to two component coupled case with the damping term, linear and parabolic density profiles, then give the Lax pair and infinitely-many conservations laws of this coupled system. The system is nonautonomous, that is, it admits a nonisospectral linear eigenvalue problem. In fact, the Darboux transformation for this kind of inhomogeneous system which is essentially different from the isospectral case, we reconstruct the Darboux transformation for this coupled Gross-Pitaevskii equation. Multi nonautonomous solitons, one breather and the first-order rogue wave are also obtained by the Darboux transformation. When $\beta >0$, the amplitudes and velocities of solitons decay exponentially as $t$ increases, otherwise, they increase exponentially as $t$ increases. Meanwhile, the real part $Re(\xi_j)$'s~$(j=1,2,3,\dots)$ of new spectral parameters determine the direction of solitions' propagation and $\alpha$ affects the localization of solitons. Choosing $Re(\xi_1)=Re(\xi_2)$, the two-soliton bound state is obtained. From nonzero background seed solutions, we construct one nonautonomous breather on curved background and find that this breather has some deformations along the direction of $t$ due to the exponential decaying term. Besides, $\beta$ determines the degree of this curved background, if we set $\beta>0$, the amplitude of the breather becomes small till being zero as $t$ increases. Through taking appropriate limit about the breather, the first-order rogue wave can be acquired. Finally, we give multi-component generalization of Gross-Pitaevskii equation and its Lax pair with nonisospectral parameter, meanwhile, Darboux transformation about this multi-component generalization is also constructed.