Journal of Nonlinear Science

, Volume 22, Issue 6, pp 1013–1040 | Cite as

Traveling Waves in Elastic Rods with Arbitrary Curvature and Torsion

  • M. J. Ablowitz
  • V. Barone
  • S. De Lillo
  • M. SommacalEmail author


The dynamic Kirchhoff equations, describing a thin elastic rod of infinite length, are considered in connection with the study of the conformations of polymeric chains. A novel special traveling wave solution that can be interpreted as a conformational soliton propagating at constant speed is obtained, featuring arbitrary non-constant curvature and torsion of the rod, in the simple case of constant cross-section, homogeneous density and elastic isotropy. This traveling wave corresponds to a specific constraint on the twist-to-bend ratio of the constant stiffness parameters, which in turn appears to be compatible with the experimental evidence for the mechanical properties of real polymeric chains. Due to such a constraint, the square of the velocity of the solitary wave is directly proportional to the bending stiffness and inversely proportional to the density and to the principal momentum of inertia of the rod. Several applications to the study of conformational changes in polymeric chains are given.


Continuum mechanics Elastic rod Dynamic Kirchhoff equations Polymeric chains Conformational soliton Protein folding 

Mathematics Subject Classification (2010)

74Axx 74Bxx 35C08 82D60 92D20 



This research was partially supported by the National Science Foundation under grant DMS-0905779.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • M. J. Ablowitz
    • 1
  • V. Barone
    • 2
    • 3
  • S. De Lillo
    • 4
    • 5
  • M. Sommacal
    • 6
    Email author
  1. 1.Applied Mathematics DepartmentUniversity of ColoradoBoulderUSA
  2. 2.Scuola Normale SuperiorePisaItaly
  3. 3.Istituto Nazionale di Fisica NucleareSezione di PisaPisaItaly
  4. 4.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly
  5. 5.Istituto Nazionale di Fisica NucleareSezione di PerugiaPerugiaItaly
  6. 6.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance

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