He, Jingsong, Cheng, Yi and Roemer, Rudolf A.. (2006) Solving bi-directional soliton equations in the KP hierarchy by gauge transformation. Journal of High Energy Physics (Online), Vol.3 . ISSN 1029-8479

Preview

PDF WRAP_Roemer_Solving-bi-drectional.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (741Kb)

Abstract

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.

Item Type:	Journal Article
Subjects:	Q Science > QC Physics
Divisions:	Faculty of Science > Centre for Scientific Computing Faculty of Science > Physics
Library of Congress Subject Headings (LCSH):	Solitons, Kadomtsev-Petviashvili hierarchy
Journal or Publication Title:	Journal of High Energy Physics (Online)
Publisher:	Institute of Physics Publishing
ISSN:	1029-8479
Date:	31 March 2006
Volume:	Vol.3
Identification Number:	10.1088/1126-6708/2006/03/103
Status:	Peer Reviewed
Access rights to Published version:	Open Access
Funder:	China, Guo jia zi ran ke xue ji jin wei yuan hui (China) [National Natural Science Foundation of China] (NSFC), University of Warwick
Grant number:	10301030 (NSFC)
References:	1. E. Date, M. Kashiwara, M. Jimbo, T. Miwa, in Nonlinear Integrable Systems- Classical and Quantum Theory, edited by M. Jimbo and T. Miwa (World Scientic, Singapore, 1983) p. 39-119. 2. Y. Ohta, J. Satsuma, D. Takahashi, T. Tokihiro: An Elementry Introduction to Sato Theory. Prog. Theor. Phys. Suppl. 94, 210-241(1988) 3. L. A. Dickey, Soliton Equations and Hamiltonian Systems (World Scintic, Singapore, 1991). 4. M. Jimbo, T. Miwa: Solitons and Innite Dimensional Lie Algebras. Publ.RIMS, Kyoto Univ.19, 943-1001(183) 5. I. M. Gelfand, L. A. Dickey, 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 S. G. Gindikin,V. W. Guillemin, A. A. Kirillov, B. Kostant, S. Sternberg,(Berlin ; Springer-Verlag, 1987.) a) P610-624, b) P625-646. 6. K. Sawada, T. Kotera, A method of for nding N-soliton solutions of the KdV and KdV-like equation. Prog. Theor. Phys.A51, 1355-1367(1974). 7. P. J. Caudrey, R. K. Dodd, J. D. Gibbon, A new hierarchy of Korteweg-de Vires equations, Proc. R. Soc. London, Ser.A351, 407-422(1976). 8. D. J. Kaup, On the sacttering problem for the cubic eigenvalue problem of the calss:xxx + 6Qx + 6R = . Stud. Appl. Math. A62, 189-216(1980). 9. B. A. Kupershmidt, A super KdV equation: an integrable system. Phys. Lett. A102, 213-215(1984). 10. J. M. Dye, A.Parker, On bidirectional fth-order nonlinear evolution equations, Lax pairs, and directionally solitary waves. J. Math. Phys. 42, 2567-2589(2001) 11. J. M. Dye, A. Parker, A bidirectional Kaup-Kupershmidt equation and directionally dependent solitons. J. Math. Phys. 43, 4921-4949(2002) 12. D. J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal,and on a new type of long stationary waves. Philos. Mag. Ser. 5, 39, 422-443(1895). 13. J. Satsuma, R. Hirota, A coupled KdV equation is one case of the four-reduction of the KP hierarchy. J. Phys. Soc. Japan 51, 3390-3397(1982) 14. N. Yajima, M. Oikawa, Formation and interaction of sinc-Langmuir solitons{inverse scattering method. Prog. Theor. Phys.56, 1719-1739(1976). 15. M. Wadati, The modied Korteweg-de Vries equation. J. Phys. Soc. Japan 32, 1681-(1972). 16. V. E. Zakharov, A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional of waves in nonlinear media, Sov. Phys. JETP.34, 62-69(1972). 17. B. G. Konopelchenko, J. Sidorenko, W. Strampp, (1 + 1)-dimensional integrable systems as symmetry constraints of (2 + 1)-dimensional systems Phys. Lett. A157, 17-21(1991). 18. Y. Cheng, Y. S. Li, The constraint of the Kadomtsev-Petviashvili equation and its special solutions. Phys. Lett.A157, 22-26(1991). 19. W. Oevel, W. Strampp, Constrained KP hierarchy and Bi-Hamiltonian structures. Commun. Math. Phys.157, 51- 81(1993). 20. Y. Cheng, Constraints of the Kadomtsev-Petviashvili hierarchy. J. Math. Phys. 33, 3774-3782(1992) 21. Y. Cheng, Modifying the KP, the nth constrained KP hierarchies and their Hamiltonian structures. Commun. Math. Phys. 171, 661-682(1995) 22. V. E. Zakharov, A. B. Shabat, A Scheme for intgerating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. Funct. Anal. Appl.8, 226-235(1974). 23. C. Verhoeven, M. Musette, Soliton solutions of two bidirectional sixth-order partial dierential equations belonging to the KP hierarchy. J. Phys. A 36, L133-L143(2003) 24. V. E. Zakharov, A. V. Mikhailov, Relativistically invariant two-dimensional models of eld theory which are integrable by means of the inverse sacttering problem method. Sov. Phys. JETP47,1017-1027(1978). 25. V. B. Matveev, M. A. Salle, Darboux Transformations and Solitons (Springer{Verlag, Berlin, 1991). 26. L. L. Chau, J. C. Shaw, H. C. Yen, Solving the KP hierarchy by gauge transformations. Commun. Math. Phys.149, 263-278(1992). 27. J. S. He, Y. S. Li, Y. Cheng, The determinant representation of the gauge transformation operators. Chin. Ann. of Math.23B, 475-486(2002). 28. A. Nakamura, A bilinear n-soliton formula for the KP equation. J. Phys. Soc.Japan.58,412-422(1989). Solving bi-directional soliton equations . . . 31 29. W. Oevel, W. Schief, "Darobux theorem and the KP hierarchy" in Application of Nonlinear Dierential Equations, edited by P. A. Clarkson(Dordrecht: Kluwer Academic Publisher, 1993)P193-206. 30. J. J. Nimmo, "Darboux transformation from reduction of the KP hierarchy ", in Nonlinear Evolution equation and Dynamical Systems , edited by V. G. Makhankov et al(Singapore: World Scientic, 1995)P168-177. 31. I. Loris, On reduced CKP equations. Inverse Problems.15, 1099-1109(1999). 32. I. Loris, R. Willox, Symmetry reductions of BKP hierarchy. J. Math. Phys.40, 1420-1431(1999). 33. W. Oevel, Darboux theorems and Wronskian formulas for integrable systems. I. Constrained KP ows. Physica A195, 533-576(1993) 34. H. Aratyn, E. Nissimov, S. Pacheva, "Constrained KP Hierarchies: Darboux-B�acklund Solutions and Additional Symmetries." Preprint (solv-int/9512008) 35. L. L. Chau, J. C. Shaw, M. H. Tu, Solving the constrained KP hierarchy by gauge transformations. J. Math. Phys. 38, 4128-4137(1997) 36. J. S. He, Y. S. Li, Y. Cheng, Two Choices of the Gauge transformation for the AKNS hierarchy through the constrained KP hierarchy. J. Math. Phys.44, 3928-3960(2003). 37. R. Hirota, Direct methods in soliton theory, in Solitons, edited by R.K.Bullough and P.J.Caudrey ,Topics in Current Physics,vol.17, 157-176(Springer,Berlin,1980) 38. C. Verhoeven, M. Musette, Grammian N-soliton solutions of a coupled KdV system. J. Phys. A 34, L721-L725(2001) 39. A. Mei, Darboux Transformations for Antisymmetric Opertor and BKP Integrable Hierarchy(in Chinese). Master Thesis, University of Science and Tehnology of China(1999). 40. V. G. Drinfeld, V. V. Sokolov, New evloution equations having (L-A)-pairs, Trudy Sem. S. L. Soboloeva, Inst. Mat. Novosibirsk 2,5-9(1981)[in Russian].
URI:	http://wrap.warwick.ac.uk/id/eprint/353

Request changes to a record

Actions (login required)

View Item