  We investigate fast transport and spin manipulation of tunable
spin-orbit-coupled Bose-Einstein condensates in a moving harmonic trap.
Motivated by the concept of "shortcuts to adiabaticity", we design inversely
the time-dependent trap position and spin-orbit coupling strength. By choosing
appropriate boundary conditions we obtain fast transport and spin flip
simultaneously. The non-adiabatic transport and relevant spin dynamics are
illustrated with numerical examples, and compared with the adiabatic transport
with constant spin-orbit-coupling strength and velocity. Moreover, the
influence of nonlinearity induced by interatomic interaction is discussed in
terms of the Gross-Pitaevskii approach, showing the robustness of the proposed
protocols. With the state-of-the-art experiments, such inverse engineering
technique paves the way for coherent control of spin-orbit-coupled
Bose-Einstein condensates in harmonic traps.
