We observed the splitting of the fluxon-flow branches in the current-voltage characteristics of serially stacked intrinsic Josephson junctions (IJJs) formed in $Bi_2$$Sr_2$$CaCu_2$$O_{ 8＋x}$ single crystals in the long-junction limit. Stacks of IJJs were sandwiched between two Au electrodes deposited on the top and the bottom of the stack using the ‘double-side cleaving technique’. In all the samples studied, the branch splitting started occurring for a dense fluxon configuration around 2 T and became more distinct in a higher magnetic field range. This observation can be explained in terms of switching between different Josephson fluxon modes in resonance with the collective plasma oscillations induced by both inductive and capacitive coupling between stacked IJJs. This is the first detailed and quantitative identification of the coherent flux-flow modes in stacked..