Il Nuovo Cimento B (1971-1996)

, Volume 111, Issue 9, pp 1135–1149 | Cite as

Binary non-linearization of Lax pairs of Kaup-Newell soliton hierarchy

  • W. -X. Ma
  • Q. Ding
  • W. G. Zhang
  • B. Q. Lu


Kaup-Newell soliton hierarchy is derived from a kind of Lax pairs different from the original ones. A binary non-linearization procedure corresponding to the Bargmann symmetry constraint is carried out for those Lax pairs. The proposed Lax pairs together with adjoint Lax pairs are constrained as a hierarchy of commutative, finite-dimensional integrable Hamiltonian systems in the Liouville sense, which also provides us with new examples of finite-dimensional integrable Hamiltonian systems. A sort of involutive solutions to the Kaup-Newell hierarchy are exhibited through the obtained finite-dimensional integrable systems and the general involutive system engendered by binary non-linearization is reduced to a specific involutive system generated by mono-non-linearization.


03.40.Kg Waves and wave propagation: general mathematical aspects 


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  1. [1]
    Ablowitz M. J., Ramani A. andSegur H.,Lett. Nuovo Cimento,23 (1978) 333;J. Math. Phys.,21 (1980) 715, 1006.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Airault H., McKean H. P. andMoser J.,Commun. Pure Appl. Math.,30 (1977) 95;Case K. M.,Proc. Natl Acad. Sci.,75 (1978) 3562;76 (1979) 1.MathSciNetADSCrossRefMATHGoogle Scholar
  3. [3]
    Cao C. W.,Chin. Quart. J. Math.,3 (1988) 90;Sci. China A,33 (1990) 528.Google Scholar
  4. [4]
    Cao C. W. andGeng X. G.,J. Phys. A,23 (1990) 4117;J. Math. Phys.,32 (1991) 2323.MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    Zeng Y. B. andLi Y. S.,J. Math. Phys.,30 (1989) 1679;Zeng Y. B.,Phys. Lett. A,160 (1991) 541.MathSciNetADSCrossRefMATHGoogle Scholar
  6. [6]
    Ma W. X. andStrampp W.,Phys. Lett. A,185 (1994) 277.MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    Ma W. X.,J. Phys. Soc. Jpn.,64 (1995) 1085;Physica A,219 (1995) 467;Binary non-linearization for the Dirac systems, to be published inChinese Annals of Math, B (solv-int/9512002).ADSCrossRefGoogle Scholar
  8. [8]
    Antonowicz M. andWojciechowski S.,Phys. Lett. A,147 (1990) 455;J. Phys A,24 (1991) 5043;J. Math. Phys.,33 (1992) 2115;Ragnisco O. andWojciechowski S.,Inverse Problems,8 (1992) 245;Blaszak M.,Phys. Lett. A,174 (1993) 85;Tondo G.,J. Phys. A,28 (1995) 5097.MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    Kaup D. J. andNewell A. C.,J. Math. Phys.,19 (1978) 798.MathSciNetADSCrossRefMATHGoogle Scholar
  10. [10]
    Tu G. Z.,J. Phys. A,22 (1989) 2375;Ma W. X.,J. Phys. A,26 (1993) 2573.MathSciNetADSCrossRefMATHGoogle Scholar
  11. [11]
    Fordy A. P. andGibbons J.,J. Math. Phys.,21 (1980) 2508;22 (1981) 1170.MathSciNetADSCrossRefMATHGoogle Scholar
  12. [12]
    Fuchssteiner B.,Nonlinear Anal TMA,3 (1979) 849;Prog. Theor. Phys.,65 (1981) 861.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Tu G. Z.,Sci. Sinica A,24 (1986) 138;J. Math. Phys.,30 (1989) 330.MATHGoogle Scholar
  14. [14]
    Tu G. Z. andMeng D. Z.,Nonlinear Evolutions, edited byJ. J. P. Leon (World Scientific, Singapore) 1988, p. 425;Acta Math. Appl. Sinica,5 (1989) 89.Google Scholar
  15. [15]
    Magri F.,Nonlinear Evolution Equations and Dynamical Systems, edited byM. Boiti,F. Pempinelli andG. Soliani (Springer-Verlag, Berlin) 1980, p. 233.CrossRefGoogle Scholar
  16. [16]
    Ma W. X.,J. Math. Phys.,33 (1992) 2464;J. Phys. A,25 (1992) 5329.MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    Arnold V. I.,Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin) 1978;Abraham R. andMarsden J.,Foundations of Mechanics, 2nd edition (Addison-Wesley, Reading, Mass.) 1978.CrossRefMATHGoogle Scholar
  18. [18]
    Liu C. P.,Appl Math.-J. Chin. Universities, Ser. A,8 (1993) 157.ADSGoogle Scholar
  19. [19]
    Ma W. X. andFuchssteiner B.,Binary non-linearization of Lax pairs, inProceedings of the International Conference of Nonlinear Physics, Gallipoli Italy (World Scientific, Singapore) 1996.Google Scholar

Copyright information

© Società Italiana di Fisica 1996

Authors and Affiliations

  • W. -X. Ma
    • 1
    • 2
  • Q. Ding
    • 1
  • W. G. Zhang
    • 3
  • B. Q. Lu
    • 1
  1. 1.Institute of MathematicsFudan UniversityShanghaiPRC
  2. 2.FB Mathematik und InformatikUniversitÄt PaderbornPaderbornGermany
  3. 3.Department of Mathematics and MechanicsChangsha Railway UniversityChangshaPRC

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