Localized buckling of a floating elastica

We study the buckling of a two-dimensional elastica floating on a bath of dense fluid, subjected to axial compression. The sinusoidal pattern predicted by the analysis of linear stability is shown to become localized above the buckling threshold. A nonlinear amplitude equation is derived for the envelope of the pattern. These results provide a simple interpretation to the wrinkle-to-fold transition reported by Pocivavsek et al. [Science 320, 912 (2008)]. An analogy with the classical problem of the localized buckling of a strut on a nonlinear elastic foundation is presented.