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

Localized buckling solutions are known to exist in the heuristic model of a n elastic strut resting on an elastic (Winkler) foundation. The primary loc alized solution emerging from the critical buckling state consists of an am plitude envelope of approximately the form of a hyperbolic secant function which modulates a fast-varying sinusoidal oscillation. In previous works su ch solutions have been tracked for the entire post-buckling regime both num erically and through a Rayleigh-Ritz approach. A very rich structure is known to exist in the subcritical Toad range but h as been proved to exist for only a certain family of reversible systems. St udies have concentrated on symmetric homoclinic solutions and the asymmetri c 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 le ss than critical. Here we uncover a new family of antisymmetric solutions a nd some asymptotic analysis suggests that the primary antisymmetric solutio n exists over the same range as does the primary symmetric solution. A pert urbation 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 numeric al solutions.