### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 373-391 |

Number of pages | 19 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 132 |

Issue number | 3 |

State | Published - Aug 1 1999 |

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### Keywords

- Amplitude equations
- Elastic rods
- Nonlinear Klein-Gordon equations
- Nonlinear Schrödinger equations

### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*132*(3), 373-391.

**Pulses, fronts and oscillations of an elastic rod.** / Lega, Joceline C; Goriely, Alain.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 132, no. 3, pp. 373-391.

}

TY - JOUR

T1 - Pulses, fronts and oscillations of an elastic rod

AU - Lega, Joceline C

AU - Goriely, Alain

PY - 1999/8/1

Y1 - 1999/8/1

N2 - 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.

AB - 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.

KW - Amplitude equations

KW - Elastic rods

KW - Nonlinear Klein-Gordon equations

KW - Nonlinear Schrödinger equations

UR - http://www.scopus.com/inward/record.url?scp=0347596686&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347596686&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0347596686

VL - 132

SP - 373

EP - 391

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3

ER -