The effectiveness of two methods for controlling the nonlinear dynamics of quantum topological vortex structures in a long Josephson junction is analysed: (a) by means of edge current injection and (b) with a stack of two long Josephson junctions. In the first case, a new, more stable and well reproducible technique for obtaining corresponding heterostructures with an additional layer of a superconducting molybdenum–rhenium alloy and a higher-quality dielectric tin-oxide layer is developed and implemented. As experimentally shown, a negative-differential-resistance region on the current– voltage characteristic arises with increasing the current injection. In the second case, the appearance of a ‘back bend’ region on a current–voltage characteristic and the transition to a resistive state with a voltage close to the gap one due to the emergence of a flux–antifluxon pair in the adjacent junction is demonstrated theoretically.