Abstract D ouble-scale perturbation analysis of a long elastic cylindrical shell under axial compression reveals a second-order non-linear differential equation for the buckle-pattern amplitude in slow-space. Numerical solution then suggests that the most easily-triggered failure mode is localized along the length. The method is extended to include mode interaction, giving three coupled second-order non-linear differential equations in slow-space. Localized solutions are again found, by combining features of the Lagrangian function with a systematic numerical search procedure. The predicted extent of the localization, about one-and-a-half axial wavelengths when fully developed, compares well with published experiments on long cylinders. Moreover, in contrast to the associated periodic solutions, “square” waves at the minimum critical load are denied ; the predominant waveform turns out to be long axially, again as seen experimentally.