ELSEVIER Physica D 105 (1997) 45-61 PHYSICA Nonlinear dynamics of filaments II. Nonlinear analysis Alain Goriely *, Michael Tabor Universi~ of Arizona, Program in Applied Mathematics, Building #89, Tucson, AZ 85721, USA Received 12 April 1996; accepted 4 December 1996 Communicated by C.K.R.T. Jones Abstract 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. Keywords: Elasticity; Kirchhoff equations; Amplitude equations 1. Introduction The study of thin rods has a long tradition in mechanics and engineering dating back to Euler and Lagrange. In recent years the study of filamentary structures has played an increasingly important role for the description of many phenomena appearing in physical, biological and chemical systems [1-14]. The Kirchhoff equations provide a well-established model to describe the dynamics of thin filaments within the approximations of linear elasticity theory. In a companion paper [ 15], we studied the stability of stationary solutions which describe finite (or infinite) rods in an equilibrium state. The parameters controlling the stability of these solutions are often taken to be the twist and/or the tension at the ends. In the preceding paper, we developed a perturbation scheme at the level of the local ortbonormal basis (the director basis) attached to the rod axis. We derived the general linear equations controlling the stability of the given stationary solutions. In particular we fully described the instability of the twisted planar ring and the straight rod. The dispersion relations relating spatial structure to growth rates in time were derived in these cases and different situations were considered. * Corresponding author. 0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PH SO 167-2789(96)00003- 1