Instability of Pulse-like Deformations of an Elastic Filament

Introduction

Bifurcations An elastic filament kept under tension and subject to increasing twist, will undergo a series of bifurcations, as illustrated in the series of photographs shown on the right. Near the threshold of the first bifurcation (which occurs between the first and second panels on the right), the dynamics of the elastic filament may be described in terms of nonlinear coupled Klein-Gordon equations [1]. These equations rule the dynamics of the slowly varying twist and slowly varying amplitude of the helical mode which becomes unstable at the bifurcation threshold.

It was shown in [2] that the coupled Klein-Gordon equations admit a two-parameter family of particular solutions, which correspond to pulse-like deformations of the filament [2]. Numerical simulations [2] (see the figure below for the evolution of a stable pulse) indicate that these solutions may be stable or unstable, depending on the speed at which they propagate, and on the frequency at which the corresponding filament rotates upon itself.

Numerical Simulation

Instability Criterion, using the Evans Function

For non-rotating filaments, one can show by means of Evans function techniques [3], that pulses are unstable if their speed lies in a prescribed interval. The Evans function [4-8] (for a review see [9]), is a function of the spectral parameter, which vanishes on the point spectrum of a linear operator. The analysis described in [3] relies on the use of a classical parity argument, which guarantees that the Evans function has to vanish at least once on the positive real axis. One can indeed show that the Evans function is real for real values of the spectral parameter, and that it is continuous on the positive real axis. Then, knowledge of the behavior of the Evans function near the origin (typically obtained by means of asymptotic expansions), together with information on the sign of the Evans function for large values of spectral parameter, leads to a sufficient condition for the existence of a zero of this function on the positive real axis. This condition in turn implies the existence of a real positive eigenvalue of the linearized operator obtained by linearizing the coupled Klein-Gordon equations about a pulse solution. This gives the instability criterion found in [3].

Spectral Stability Criterion, using Hamiltonian Techniques

One can obtain a more general instability criterion, which applies to both rotating and non-rotating filaments, by using Hamiltonian techniques. The coupled Klein-Gordon equations are indeed Hamiltonian [2], and general results [10-11] for the orbital stability of solutions to Hamiltonian systems with symmetries can be found in the literature. The theorems of [10-11] however do not directly apply here because the continuous spectrum of the linearization about a pulse solution is the whole imaginary axis, and therefore contains the origin. One can however modify the results of [10-11] in order to obtain a necessary and sufficient condition for the spectral stability of both types of pulses [12]. The instability criterion hence obtained generalizes the results of [3].

PowerPoint Presentation

Instability of Local Deformations of an Elastic Filament

References

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  9. B. Sandstede, Stability of travelling waves, in Handbook of Dynamical Systems II: Towards Applications, pp. 983-1055, B. Fielder Ed., Elsevier, 2002.
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Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No DMS0075827.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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