We present a system of equations of motion for an array of fluxons trapped in a stack of parallel tunnel-coupled long Josephson junctions driven by a uniformly distributed ac bias current. Detailed numerical simulations yield the dependence of the critical amplitude of the drive required to destroy the fluxon string on the driving frequency. In the strongly damped system, this dependence is rather trivial, but in the absence of dissipation it exhibits deep minima, which can be associated with nonlinear resonances. Near the resonance points, the critical amplitude of the dynamical (ac) drive may be lower than the amplitude of the static (dc) drive for the same system. The problem considered here may find applications for detecting trapped fluxon arrays in high-temperature superconductors.