Abstract
We give three nonlinear partial differential equations which are associated with binormal motions of constant torsion curves in Minkowski 3-space. We also give Bäcklund transformations for these equations, as well as for surfaces swept out by related moving curves. As applications, from some trivial binormal motions we construct some new binormal motions.
Similar content being viewed by others
References
Chou K. S., Qu C. Z.: Integrable equations arising from motions of plane curves. Physica D 162, 9–33 (2002)
Chou K. S., Qu C. Z.: Integrable equations arising from motions of plane curves II. J. Nonlinear Sci. 13, 487–517 (2003)
Ding Q.: Schrödinger flows, binormal motion for curves and the second AKNS-hierarchies. Chaos, Solitons and Fractals 21, 669–677 (2004)
Dubrovin B. A., Fomenko A. T., Novikov S. P.: Modern geometry methods and applications: part I. The geometry of surfaces, transformation groups, and fields, 2nd Edition. Springer-Verlag, New York (1999)
Hasimoto H.: A soliton on a vortex filement. J. Fluid Mech. 51, 477–485 (1972)
Lakshmanan M., Ruijgrok T.W., Thompson C.J.: On the dynamics of a continuum spin system. Phys. A 84, 577–590 (1976)
Lamb G. L.: Bäcklund transformations for certain nonlinear evolution equations. J. Math. Phys. 15, 2157–2165 (1974)
R. López, Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Spaces, arXiv:0810.3351vl[math.DG] 18 Oct. 2008.
Nakayama K., Segur H., Wadati M.: Integrability and the motion of curves, Phys. Rev. Lett. 69, 2603–2606 (1992)
Nakayama K., Wadati M.: Motion of curves in the plane. J. Phys. Soc. Japan 62, 473–479 (1993)
B. O’Neill: Semi-Riemannian Geometry. Academic Press, New York (1983)
C. Rogers and W. K. Schief, Bäcklund and Darboux transformations. Geometry and Modern Applications in Soliton Theory, Cambridge University Press, Cambridge, 2002.
Schief W.K., Rogers C.: Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces, Proc. R. Soc. Lond. A 445, 3163–3188 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by the NSF of China (11101352 and 11071208) and the Natural Science Foundation of Anhui Education Committee (KJ2011D125).
Rights and permissions
About this article
Cite this article
Xu, C., Cao, X. Nonlinear partial differential equations associated with binormal motions of constant torsion curves in Minkowski 3-space. Arch. Math. 99, 481–492 (2012). https://doi.org/10.1007/s00013-012-0443-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-012-0443-6