Theoretical and Mathematical Physics

, Volume 149, Issue 1, pp 1324–1337 | Cite as

Dressing chain for the acoustic spectral problem

  • V. E. Adler
  • A. B. Shabat
Article

Abstract

We study iterations of the Darboux transformation for the generalized Schrödinger operator and consider applications to the Dym and Camassa-Holm equations.

Keywords

Darboux transformation dressing chain Dym equation Camassa-Holm equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Kruskal, “Nonlinear wave equations,” in: Dynamical Systems, Theory and Applications (Lect. Notes Phys., Vol. 38, J. Moser, ed.), Springer, Heidelberg (1975), pp. 310–354.Google Scholar
  2. 2.
    B. Fuchssteiner and A. S. Fokas, Phys. D, 4, 47–66 (1981); R. Camassa and D. D. Holm, Phys. Rev. Lett., 71, 1661–1664 (1993).MathSciNetCrossRefADSGoogle Scholar
  3. 3.
    M. G. Krein, Dokl. Akad. Nauk SSSR, 82, 669–672 (1952).MATHMathSciNetGoogle Scholar
  4. 4.
    F. Calogero and A. Degasperis, Spectral Transforms and Solitons, North-Holland, Amsterdam (1982).Google Scholar
  5. 5.
    R. Beals, D. Sattinger, and J. Szmigielski, Adv. Math., 140, 190–206 (1998).MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Constantin, Proc. Roy. Soc. London A, 457, 953–970 (2001).MATHMathSciNetADSCrossRefGoogle Scholar
  7. 7.
    L. A. Dmitrieva, Phys. Lett. A, 182, 65–70 (1993); J. Phys. A, 26, 6005–6020 (1993); A. Constantin and H. P. McKean, Comm. Pure Appl. Math., 52, 949–982 (1999); V. S. Novikov, JETP Letters, 72, 153–157 (2000); M. S. Alber and Yu. N. Fedorov, Inverse Problems, 17, 1017–1042 (2001); M. S. Alber, R. Camassa, Yu. N. Fedorov, D. D. Holm, and J. E. Marsden, Comm. Math. Phys., 221, 197–227 (2001); R. Camassa and A. I. Zenchuk, Phys. Lett. A, 281, 26–33 (2001).MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    J. Schiff, Phys. D, 121, 24–43 (1998); A. N. W. Hone, J. Phys. A, 32, L307–L314 (1999); R. I. Ivanov, Phys. Lett. A, 345, 112–118 (2005); nlin.SI/0507005 (2005).MATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    M. M. Crum, Quart. J. Math. Oxford 2, 6, 121–127 (1995).MathSciNetGoogle Scholar
  10. 10.
    G. L. Lamb Jr., Elements of Soliton Theory, Wiley, New York (1980).MATHGoogle Scholar
  11. 11.
    A. B. Shabat, Inverse Problems, 8, 303–308 (1992); A. P. Veselov and A. B. Shabat, Funct. Anal. Appl., 27, 81–96 (1993).MATHMathSciNetCrossRefADSGoogle Scholar
  12. 12.
    A. B. Shabat and R. I. Yamilov, Leningr. Math. J., 2, 377–400 (1990).MathSciNetGoogle Scholar
  13. 13.
    R. H. Heredero, J. Nonlinear Math. Phys., 12, 567–585 (2005).MATHMathSciNetADSGoogle Scholar
  14. 14.
    R. I. Yamilov, Theor. Math. Phys., 85, 1269–1275 (1990).MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    R. Beals, M. Rabelo, and K. Tenenblat, Stud. Appl. Math., 81, 125–151 (1989); T. Schäfer and C. E. Wayne, Phys. D, 196, 90–105 (2004).MATHMathSciNetGoogle Scholar
  16. 16.
    G. Falqui, “On a Camassa-Holm type equation with two dependent variables,” nlin.SI/0505059 (2005); H. Aratyn, J. F. Gomes, and A. H. Zimerman, J. Phys. A, 39, 1099–1114 (2006).Google Scholar
  17. 17.
    A. Degasperis and M. Procesi, “Asymptotic integrability,” in: Symmetry and Perturbation Theory (Proc. 2nd Intl. Workshop, Rome, Italy, December 16–22, 1998, A. Degasperis and G. Gaeta, eds.), World Scientific, Singapore (1999), pp. 23–37; A. Degasperis, D. D. Holm, and A. N. W. Hone, Theor. Math. Phys., 133, 170–183 (2002).Google Scholar
  18. 18.
    V. E. Adler, V. G. Marikhin, and A. B. Shabat, Theor. Math. Phys., 129, 1448–1465 (2001).MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    V. G. Dubrovsky and B. G. Konopelchenko, J. Phys. A, 27, 4619–4628 (1994); L. A. Dmitrieva and M. A. Khlabystova, Phys. Lett. A, 237, 369–380 (1998).MATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. E. Adler
    • 1
  • A. B. Shabat
    • 1
  1. 1.Landau Institute for Theoretical PhysicsRASChernogolovkaRussia

Personalised recommendations