Quick Search:
Author: Title/Abstract: Vol./No: Page:

Prog. Theor. Phys. Vol. 51 No. 3 (1974) pp. 703-716

[ Full Text PDF : FREE ACCESS (1053K) ]

On the Toda Lattice. II

— Inverse-Scattering Solution —

H. Flaschka

Department of Mathematics, The University of Arizona, Tucson, Arizona 85721

(Received September 17, 1973)

Abstract:

An inverse-scattering method is developed for the solution of the exponential lattice of Toda. As application of this method, the general N-soliton formula is derived, and constants of the motion are expressed in terms of the scattering data.


URL : http://ptp.ipap.jp/link?PTP/51/703/
DOI : 10.1143/PTP.51.703

[ Full Text PDF : FREE ACCESS (1053K) ] Citation:


References:

  1. M. Toda, Prog. Theor. Phys. Suppl. No. 45 (1970), 174[PTP].
  2. M. Toda, J. Phys. Soc. Jpn. 22 (1967), 431[JPSJ]; ibid. 23 (1967), 501[JPSJ].
  3. J. Ford, S. D. Stoddard and J. S. Turner, Prog. Theor. Phys., to appear.
  4. M. Hènon, to appear in Phys. Rev.
  5. H. Flaschka, to appear in Phys. Rev.
  6. M. Toda and M. Wadati, J. Phys. Soc. Jpn. 34 (1973), 18[JPSJ].
  7. C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Phys. Rev. Lett. 19 (1967), 1095[APS].
  8. V. E. Zakharov and L. D. Faddeev, Funk. Anal. i evo Priloz. 5 (1971), 18.
  9. M. Wadati, J. Phys. Soc. Jpn. 34 (1973), 1289[JPSJ].
  10. V. E. Zakharov and A. B. Shabat, Sov. Phys. -JETP 34 (1972), 62.
  11. M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Phys. Rev. Lett. 30 (1973), 1262[APS]; ibid. 31 (1973), 125[APS].
  12. R. Hirota, J. Phys. Soc. Jpn. 35 (1973), 286[JPSJ].
  13. K. M. Case and M. Kac, J. Math. Phys. 14 (1973), 594[CrossRef].
  14. K. M. Case, J. Math. Phys. 14 (1973), 916[CrossRef].
  15. L. D. Faddeev, Am. Math. Soc. Translations 65 (1967), 139.
  16. P. D. Lax, Comm. Pure Appl. Math 21 (1968), 467.
  17. For a detailed account of the spectral theory of difference operators, see Ju. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators, AMS Transl. of Math. Monographs, v. 17, Providence, R. I. 1968.
  18. V. de Alfaro and T. Regge, Potential Scattering (North-Holland, Amsterdam 1965).
  19. I. M. Gel'fand and B. M. Levitan, Am. Math. Soc. Translations 1 (1955), 253.
  20. Z. S. Agranovic and V. A. Marčenko, The Inverse Problem of Scattering Theory (Gordon and Breach, New York, 1963).
  21. M. J. Ablowitz and A. C. Newell, J. Math. Phys. 14 (1973), 1277[CrossRef].
  22. M. Wadati and M. Toda, J. Phys. Soc. Jpn. 32 (1972), 1403[JPSJ].
  23. V. S. Buslaev and L. D. Faddeev, DAN 132 (1960), 13.
  24. I. C. Percival, Proc. Phys. Soc. 80 (1962), 1290.
  25. V. S. Buslaev, Topics in Mathematical Physics, ed. M. Sh. Birman, v.l, (Leningrad 1966. Engl. Transl. Plenum Press, 1967).
  26. I. M. Gel'fand and N. Ja, Vilenkin, Generalized Functions, v. 4, (Academic Press, New York, 1965).
  27. A. I. Markushevich, Theory of Functions of a Complex Variable, v. 2, (Prentice-Hall, Englewood Cliffs, N. J. 1965).

Citing Article(s) :

  1. Journal of the Physical Society of Japan 59 (1990) pp. 1163-1172 :
    Experiment on Solitary Wave in a Symmetric Electric Circuit
    Shingo Ishiwata, Shinsuke Watanabe and Hiroshi Tanaca
  2. Journal of the Physical Society of Japan 66 (1997) pp. 2608-2617 :
    Tri-Hamiltonian Structure and Complete Integrability of Volterra Model
    Takayuki Tsuchida, Yasumasa Kajinaga and Miki Wadati
  3. Journal of the Physical Society of Japan 74 (2005) pp. 1973-1979 :
    N Soliton Solution of Integrable Differential-Difference Equations
    Masashi Ito, Kimiaki Konno and Hiroshi Kakuhata
  4. Progress of Theoretical Physics Vol. 53 No. 2 (1975) pp. 419-436 :
    Relationships among Inverse Method, Bäcklund Transformation and an Infinite Number of Conservation Laws
    Miki Wadati, Heiji Sanuki and Kimiaki Konno
  5. Progress of Theoretical Physics Vol. 54 No. 1 (1975) pp. 72-78 :
    Finding Eigenvalue Problems for Solving Nonlinear Evolution Equations
    D. J. Kaup
  6. Progress of Theoretical Physics Vol. 55 No. 2 (1976) pp. 438-456 :
    Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions
    H. Flaschka and D. W. McLaughlin
  7. Progress of Theoretical Physics Vol. 55 No. 2 (1976) pp. 457-465 :
    Analogue of Inverse Scattering Theory for the Discrete Hill's Equation and Exact Solutions for the Periodic Toda Lattice
    Etsuro Date and Shunichi Tanaka
  8. Progress of Theoretical Physics Supplement No.59 (1976) pp. 1-35 :
    Development of the Theory of a Nonlinear Lattice
    Morikazu Toda
  9. Progress of Theoretical Physics Supplement No.59 (1976) pp. 36-63 :
    Transformation Theories for Nonlinear Discrete Systems
    Miki Wadati
  10. Progress of Theoretical Physics Supplement No.59 (1976) pp. 64-100 :
    A Variety of Nonlinear Network Equations Generated from the Bäcklund Transformation for the Toda Lattice
    Ryogo Hirota and Junkichi Satsuma
  11. Progress of Theoretical Physics Supplement No.59 (1976) pp. 107-125 :
    Periodic Multi-Soliton Solutions of Korteweg-de Vries Equation and Toda Lattice
    Etsuro Date and Shunichi Tanaka