Frontiers of Mathematics in China

, Volume 8, Issue 5, pp 1017–1029 | Cite as

Integrable discretizations of the Dym equation

  • Bao-Feng Feng
  • Jun-ichi Inoguchi
  • Kenji Kajiwara
  • Ken-ichi Maruno
  • Yasuhiro Ohta
Research Article


Integrable discretizations of the complex and real Dym equations are proposed. N-soliton solutions for both semi-discrete and fully discrete analogues of the complex and real Dym equations are also presented.


Dym equation integrable discretization N-soliton solution 


37K10 35Q51 39A14 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dmitrieva L A. N-loop solitons and their link with the complex Harry Dym equation. J Phys A: Math Gen, 1994, 27: 8197–8205MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Feng B-F, Inoguchi J, Kajiwara K, Maruno K, Ohta Y. Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves. J Phys A: Math Theor, 2011, 44: 395201MathSciNetCrossRefGoogle Scholar
  3. 3.
    Feng B-F, Maruno K, Ohta Y. A self-adaptive moving mesh method for the Camassa-Holm equation. J Comp Appl Math, 2010, 235: 229–243MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Goldstein R E, Petrich D M. The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane. Phys Rev Lett, 1991, 67: 3203–3206MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Hereman W, Banerjee P P, Chatterjee M R. Derivation and implicit solution of the Harry Dym equation and its connections with the Korteweg-de Vries equation. J Phys A: Math Gen, 1989, 22: 241–255MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Inoguchi J, Kajiwara K, Matsuura N, Ohta Y. Motion and Bäcklund transformations of discrete plane curves. Kyushu J Math, 2012, 66: 303–324MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Inoguchi J, Kajiwara K, Matsuura N, Ohta Y. Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves. J Phys A: Math Theor, 2012, 45: 045206MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ishimori Y. A relationship between the Ablowitz-Kaup-Newell-Segur and Wadati-Konno-Ichikawa schemes of the inverse scattering method. J Phys Soc Jpn, 1982, 51: 3036–3041MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kadanoff L P. Exact solutions for the Saffman-Taylor problem with surface tension. Phys Rev Lett, 1990, 65: 2986–2988CrossRefGoogle Scholar
  10. 10.
    Kawamoto S. An exact transformation from the Harry Dym equation to the modified KdV equation. J Phys Soc Jpn, 1985, 54: 2055–2056MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kruskal M D. Nonlinear wave equations. In: Moser J, ed. Dynamical Systems, Theory and Applications. Lecture Note in Physics, Vol 38. New York: Springer-Verlag, 1975, 310–354CrossRefGoogle Scholar
  12. 12.
    Ohta Y, Maruno K, Feng B -F. An integrable semi-discretization of the Camassa-Holm equation and its determinant solution. J Phys A: Math Theor, 2008, 41: 355205MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rogers C, Wong P. On reciprocal Bäcklund transformations of inverse scattering schemes. Physica Scripta, 1984, 330: 10–14MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Wadati M, Konno K, Ichikawa Y H. New integrable nonlinear evolution equations. J Phys Soc Jpn, 1979, 47: 1698–1700MathSciNetCrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Bao-Feng Feng
    • 1
  • Jun-ichi Inoguchi
    • 2
  • Kenji Kajiwara
    • 3
  • Ken-ichi Maruno
    • 1
  • Yasuhiro Ohta
    • 4
  1. 1.Department of MathematicsThe University of Texas-Pan AmericanEdinburgUSA
  2. 2.Department of Mathematical SciencesYamagata UniversityYamagataJapan
  3. 3.Institute of Mathematics for IndustryKyushu UniversityFukuokaJapan
  4. 4.Department of MathematicsKobe UniversityRokko, KobeJapan

Personalised recommendations