Controlled nonautonomous matter–wave solitons in spinor Bose–Einstein condensates with spatiotemporal modulation

To study controlled evolution of nonautonomous matter–wave solitons in spinor Bose–Einstein condensates with spatiotemporal modulation, we focus on a system of three coupled Gross–Pitaevskii (GP) equations with space–time-dependent external potentials and temporally modulated gain/loss distributions. An integrability condition and a nonisospectral Lax pair for the coupled GP equations are obtained. Using it, we derive an infinite set of dynamical invariants, the first two of which are the mass and momentum. The Darboux transform is used to generate one- and two-soliton solutions. Under the action of different external potentials and gain/loss distributions, various solutions for controlled nonautonomous matter–wave solitons of both ferromagnetic and polar types are obtained, such as self-compressed, snake-like and stepwise solitons, and as well as breathers. In particular, the formation of states resembling rogue waves, under the action of a sign-reversible gain–loss distribution, is demonstrated too. Shape-preserving and changing interactions between two nonautonomous matter–wave solitons and bound states of solitons are addressed too. In this context, spin switching arises in the polar-ferromagnetic interaction. Stability of the nonautonomous matter–wave solitons is verified by means of systematic simulations of their perturbed evolution.