Skip to main content
SpringerLink
Log in
Book cover

Mathematical Progress in Expressive Image Synthesis III pp 137–149Cite as

dNLS Flow on Discrete Space Curves

  • Conference paper
  • First Online:

  • 401 Accesses

Part of the book series: Mathematics for Industry ((MFI,volume 24))

Abstract

The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schrödinger equation (NLS). In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schrödinger equation (dNLS). We also present explicit formulas for both NLS and dNLS flows in terms of the \(\tau \) function of the 2-component KP hierarchy.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    The N -soliton solution for the tangent vector has been constructed by using the bilinear formalism in [5].

References

  1. M.J. Ablowitz, J.F. Ladik, A nonlinear difference scheme and inverse scattering. Stud. Appl. Math. 55, 213–229 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. M.J. Ablowitz, J.F. Ladik, On the solution of a class of nonlinear partial difference equations. Stud. Appl. Math. 57, 1–12 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Doliwa, P.M. Santini, Integrable dynamics of a discrete curve and the Ablowitz-Ladik hierarchy. J. Math. Phys. 36, 1259–1273 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Doliwa, P.M. Santini, The integrable dynamics of a discrete curve, in Symmetries and Integrability of Difference Equations, vol. 9, CRM Proceedings & Lecture Notes, ed. by D. Levi, L. Vinet, P. Winternitz (American Mathematical Society, Providence, 1996), pp. 91–102

    Google Scholar 

  5. Y. Fukumoto, T. Miyazaki, N-Solitons on a curved vortex filament. J. Phys. Soc. Jpn. 55, 4152–4155 (1986)

    Article  Google Scholar 

  6. H. Hasimoto, Soliton on a vortex filament. J. Fluid Mech. 51, 477–485 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Hisakado, M. Wadati, Moving discrete curve and geometric phase. Phys. Lett. A 214, 252–258 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. T. Hoffmann, On the equivalence of the discrete nonlinear Schrödinger equation and the discrete isotropic Heisenberg magnet. Phys. Lett. A 265, 62–67 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. T. Hoffmann, Discrete Hashimoto surfaces and a doubly discrete smoke-ring flow, in Discrete Differential Geometry, vol. 38, Oberwolfach Seminars, ed. by A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler (Birkhäuser, Basel, 2008), pp. 95–115

    Chapter  Google Scholar 

  10. J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta, Motion and Bäcklund transformations of discrete plane curves. Kyushu J. Math. 66, 303–324 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta, Discrete mKdV and discrete sine-Gordon flows on discrete space curves. J. Phys. A: Math. Theor. 47, 235202 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta, Discrete models of isoperimetric deformation of plane curves, in Mathematical Progress in Expressive Image Synthesis, vol. 4, Mathematics for Industry, ed. by K. Anjyo (Springer, Tokyo, 2014), pp. 111–122

    Chapter  Google Scholar 

  13. G.L. Lamb, Solitons on moving space curves. J. Math. Phys. 18, 1654–1659 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  14. N. Matsuura, Discrete KdV and discrete modified KdV equations arising from motions of planar discrete curves. Int. Math. Res. Not. 2012, 1681–1698 (2012)

    MathSciNet  MATH  Google Scholar 

  15. K. Nakayama, Elementary vortex filament model of the discrete nonlinear Schrödinger equation. J. Phys. Soc. Jpn. 76, 074003 (2007)

    Article  Google Scholar 

  16. K. Nakayama, H. Segur, M. Wadati, Integrability and the motions of curves. Phys. Rev. Lett. 69, 2603–2606 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. U. Pinkall, B. Springborn, S. Weißmann, A new doubly discrete analogue of smoke ring flow and the real time simulation of fluid flow. J. Phys. A: Math. Theor. 40, 12563–12576 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Tsujimoto, Discretization of integrable systems, in Applied Integrable Systems, ed. by Y. Nakamura (Shokabo, Tokyo, 2000), pp. 1–52 (In Japanese)

    Google Scholar 

  19. S. Weißmann, U. Pinkall, Real-time interactive simulation of smoke using discrete integrable vortex filaments, in VRIPhys: Workshop on Virtual Reality Interaction and Physical Simulations, ed. by H. Prautzsch, et al. (The Eurographics Association, Aire-la-Ville, 2009), pp. 1–10

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Promotion of Educational Innovation, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama, 337-8570, Japan

    Sampei Hirose

  2. Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan

    Jun-ichi Inoguchi

  3. Institute of Mathematics for Industry, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Kenji Kajiwara

  4. Department of Applied Mathematics, Fukuoka University, Nanakuma 8-19-1, Fukuoka, 814-0180, Japan

    Nozomu Matsuura

  5. Department of Mathematics, Kobe University, Rokko, Kobe, 657-8501, Japan

    Yasuhiro Ohta

Authors
  1. Sampei Hirose
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun-ichi Inoguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kenji Kajiwara
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nozomu Matsuura
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yasuhiro Ohta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kenji Kajiwara .

Editor information

Editors and Affiliations

  1. Grad. School of Info. Sci. and Tech, Hokkaido University, Sapporo, Hokkaido, Japan

    Yoshinori Dobashi

  2. Kyushu University, Fukuoka, Japan

    Hiroyuki Ochiai

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media Singapore

About this paper

Cite this paper

Hirose, S., Inoguchi, Ji., Kajiwara, K., Matsuura, N., Ohta, Y. (2016). dNLS Flow on Discrete Space Curves. In: Dobashi, Y., Ochiai, H. (eds) Mathematical Progress in Expressive Image Synthesis III. Mathematics for Industry, vol 24. Springer, Singapore. https://doi.org/10.1007/978-981-10-1076-7_14

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-1076-7_14

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-1075-0

  • Online ISBN: 978-981-10-1076-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation