Integrable discretizations of the Dym equation

Abstract

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.

Keywords

Dym equation / integrable discretization / N-soliton solution

References

[1] Dmitrieva L A. N-loop solitons and their link with the complex Harry Dym equation. J Phys A: Math Gen, 1994, 27: 8197-8205

[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: 395201

[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: 229-243

[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: 3203-3206

[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: 241-255

[6] Inoguchi J, Kajiwara K, Matsuura N, Ohta Y. Motion and Bäcklund transformations of discrete plane curves. Kyushu J Math, 2012, 66: 303-324

[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: 045206

[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: 3036-3041

[9] Kadanoff L P. Exact solutions for the Saffman-Taylor problem with surface tension. Phys Rev Lett, 1990, 65: 2986-2988

[10] Kawamoto S. An exact transformation from the Harry Dym equation to the modified KdV equation. J Phys Soc Jpn, 1985, 54: 2055-2056

[11] Kruskal M D. Moser J. Nonlinear wave equations. Dynamical Systems, Theory and Applications, 1975, New York: Springer-Verlag 310 354

[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: 355205

[13] Rogers C, Wong P. On reciprocal Bäcklund transformations of inverse scattering schemes. Physica Scripta, 1984, 30: 10-14

[14] Wadati M, Konno K, Ichikawa Y H. New integrable nonlinear evolution equations. J Phys Soc Jpn, 1979, 47: 1698-1700