Investigate the dynamics of the matching-mismatching (C-IC) transition in a one-dimensional Frenkel-Kontorova chain. The disproportionate ratio between the period of the trapping potential and the average interparticle distance causes atomic rearrangements in configurations where the atoms occupy positions deviated from the potential minimum. We show that transitions in non-equilibrium dynamics feature a mechanism of discrete injection of atomic mismatches stepping in from the ends of the chain. Moreover, when the system is quenched near the boundary of the C-IC transition, quantum fluctuations can have a large impact on the dynamics, resulting in a response characterized by the superposition of matched and unmatched configurations of the chain. occurs. The mismatch motion can be precisely mapped to the dynamics of solitons moving along the chain, allowing us to investigate the discrete injection and nucleation mechanisms of the topological structure of our model. We also provide a concrete perspective for implementing the dynamics of these topological defects in trapped ion experiments.
