Pulses, fronts and oscillations of an elastic rod

Two coupled nonlinear Klein-Gordon equations modeling the three-dimensional dynamics of a twisted elastic rod near its first bifurcation threshold are analyzed. First, it is shown that these equations are Hamiltonian and that they admit a two-parameter family of traveling wave solutions. Second, special solutions corresponding to simple deformations of the elastic rod are considered. The stability of such configurations is analyzed by means of two coupled nonlinear Schrödinger equations, which are derived from the nonlinear Klein-Gordon equations in the limit of small deformations. In particular, it is shown that periodic solutions are modulationally unstable, which is consistent with the looping process observed in the writhing instability of elastic filaments. Third, numerical simulations of the nonlinear Klein-Gordon equations suggesting that traveling pulses are stable, are presented.