Effects of exponentially small terms in the perturbation approach to localized buckling

Localized buckling solutions are known to exist in the heuristic model of an elastic strut resting on an elastic (Winkler) foundation. The primary localized solution emerging from the critical buckling state consists of an amplitude envelope of approximately the form of a hyperbolic secant function which modulates a fast-varying sinusoidal oscillation. In previous works such solutions have been tracked for the entire post-buckling regime both numerically and through a Rayleigh-Ritz approach. A very rich structure is known to exist in the subcritical Toad range but has been proved to exist for only a certain family of reversible systems. Studies have concentrated on symmetric homoclinic solutions and the asymmetric solutions which bifurcate from these solution paths. The primary solution is known to exist for the entire subcritical parameter range and all other symmetric and associated asymmetric solutions exist strictly for values less than critical. Here we uncover a new family of antisymmetric solutions and some asymptotic analysis suggests that the primary antisymmetric solution exists over the same range as does the primary symmetric solution. A perturbation approach can be used to describe the bifurcation hierarchy for the novel antisymmetric forms. We illustrate a unified approach which is able to predict the circumstances under which non-divergent localized solutions are possible and the results of the analysis are compared with some numerical solutions.