Nonlinear dynamics of filaments

The Kirchhoff equations provide a well-established framework to study the statics and dynamics of thin elastic filaments. The study of static solutions to these equations has a long history and provides the basis for many investigations, both past and present, of the configurations taken by filaments subject to various external forces and boundary conditions. Here we review recently developed techniques involving linear and nonlinear analyses that enable one to study, in some detail, the actual dynamics of filament instabilities and the localized structures that can ensue. By introducing a novel arc-length preserving perturbation scheme a linear stability analysis can be performed which, in turn, leads to dispersion relations that provide the selection mechanism for the shape of an unstable filament. These dispersion relations provide the starting point for nonlinear analysis and the derivation of new amplitude equations which describe the filament dynamics above the instability threshold. Here we will mainly be concerned with the analysis of rods of circular cross-sections and survey the behavior of rings, rods, helices and show how these results lead to a complete dynamical description of filament buckling.