Location of Repository

Solving bi-directional soliton equations in the KP hierarchy by gauge transformation

By Jingsong He, Yi Cheng and Rudolf A. Roemer


We present a systematic way to construct solutions of the (n = 5)-reduction of the BKP and CKP hierarchies from the general τ function τn+k of the KP hierarchy. We obtain the one-soliton, two-soliton, and periodic solution for the bi-directional Sawada-Kotera (bSK), the bi-directional Kaup-Kupershmidt (bKK) and also the bi-directional Satsuma-Hirota (bSH) equation. Different solutions such as left- and right-going solitons are classified according to the symmetries of the 5th roots of eiε. Furthermore, we show that the soliton solutions of the n-reduction of the BKP and CKP hierarchies with n = 2j+1, j = 1,2,3,..., can propagate along j directions in the 1+1 space-time domain. Each such direction corresponds to one symmetric distribution of the nth roots of eiε. Based on this classification, we detail the existence of two-peak solitons of the n-reduction from the Grammian τ function of the sub-hierarchies BKP and CKP. If n is even, we again find two-peak solitons. Last, we obtain the ``stationary" soliton for the higher-order KP hierarchy.\u

Topics: QC
Publisher: Institute of Physics Publishing
Year: 2006
OAI identifier: oai:wrap.warwick.ac.uk:353

Suggested articles



  1. 1
  2. 2
  3. 2k1k2
  4. 2k1k2 cos("1 + "2) 32 Jingsong
  5. 2k1k2 sin"1 sin"2 +2k1k2 sin"2 doi
  6. 2k2
  7. 4k1k2
  8. A bidirectional Kaup-Kupershmidt equation and directionally dependent solitons. doi
  9. A bilinear n-soliton formula for the KP equation. doi
  10. A coupled KdV equation is one case of the four-reduction of the KP hierarchy. doi
  11. A method of for doi
  12. A new hierarchy of Korteweg-de Vires equations, doi
  13. A Scheme for intgerating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. doi
  14. A super KdV equation: an integrable system. doi
  15. A.Parker, On bidirectional nonlinear evolution equations, Lax pairs, and directionally solitary waves. doi
  16. (1987). a) Fractional powers of operators and Hamiltonian systems; b)A family of Hamiltonian structures related to nonlinear integrable partial dierential eqations, in I.M.Gelfand Collected Papers vol.I, edited by doi
  17. ck
  18. Constrained KP Hierarchies: Darboux-B acklund Solutions and Additional Symmetries." doi
  19. Constrained KP hierarchy and Bi-Hamiltonian structures. doi
  20. Constraints of the Kadomtsev-Petviashvili hierarchy. doi
  21. cos("1
  22. cos"1
  23. cos"2
  24. cos2 "2 (D.6) Communicated by Prof. P. Constantin
  25. cos2"1
  26. d2
  27. Darboux theorems and Wronskian formulas for integrable systems. I. Constrained KP ows. doi
  28. (1995). Darboux transformation from reduction of the KP hierarchy ",
  29. (1991). Darboux Transformations and Solitons (Springer{Verlag, doi
  30. Darboux Transformations for Antisymmetric Opertor and BKP Integrable Hierarchy(in Chinese). Master Thesis,
  31. (1993). Darobux theorem and the KP hierarchy" in Application of Nonlinear Dierential Equations,
  32. Direct methods in soliton theory, in Solitons, edited by R.K.Bullough and P.J.Caudrey ,Topics in Current Physics,vol.17, doi
  33. Exact theory of two-dimensional self-focusing and one-dimensional of waves in nonlinear media,
  34. Formation and interaction of sinc-Langmuir solitons{inverse scattering method. doi
  35. Grammian N-soliton solutions of a coupled KdV system. doi
  36. k1k2
  37. k2
  38. Modifying the KP, the nth constrained KP hierarchies and their Hamiltonian structures. doi
  39. New evloution equations having (L-A)-pairs,
  40. On reduced CKP equations. doi
  41. On the change of form of long waves advancing in a rectangular canal,and on a new type of long stationary waves. doi
  42. On the sacttering problem for the cubic eigenvalue problem of the calss:xxx
  43. Relativistically invariant two-dimensional models of theory which are integrable by means of the inverse sacttering problem method.
  44. sin"1
  45. sin"2
  46. sin2"2
  47. (1991). Soliton Equations and Hamiltonian Systems (World Scinti doi
  48. Soliton solutions of two bidirectional sixth-order partial dierential equations belonging to the KP hierarchy. doi
  49. Solitons and In Dimensional Lie Algebras. Publ.RIMS, Kyoto Univ.19, doi
  50. Solving the constrained KP hierarchy by gauge transformations. doi
  51. Solving the KP hierarchy by gauge transformations. doi
  52. Symmetry reductions of BKP hierarchy. doi
  53. Take
  54. The constraint of the Kadomtsev-Petviashvili equation and its special solutions. doi
  55. The determinant representation of the gauge transformation operators. doi
  56. The modi Korteweg-de Vries equation. doi
  57. Tokihiro: An Elementry Introduction to Sato Theory. doi
  58. Two Choices of the Gauge transformation for the AKNS hierarchy through the constrained KP hierarchy. doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.