Nonlinear dynamics of filaments II. Nonlinear analysis

The linear stability analysis of the full time-dependent Kirchhoff equations for elastic filaments gives precise information about possible dynamical instabilities. The associated dispersion relations derived in the preceding paper provides the selection mechanism for the shapes selected by highly unstable filaments. Here we perform a nonlinear analysis and derive new amplitude equations which describe the dynamics above the instability threshold. The straight filament is studied in detail and the motion is shown to be described by a pair of nonlinear Klein-Gordon equations which couple the local deformation amplitude to the twist density. Of particular interest is the effect of boundary conditions on the instability threshold. It is shown that with suitable choice of boundary conditions the threshold of instability is delayed. We also show the existence of pulse-like and front-like traveling wave solutions.