Theoretical and Mathematical Physics

, Volume 152, Issue 2, pp 1132–1145 | Cite as

Cylindrical Kadomtsev-Petviashvili equation: Old and new results

  • C. KleinEmail author
  • V. B. Matveev
  • A. O. Smirnov


We review results on the cylindrical Kadomtsev-Petviashvili (CKP) equation, also known as the Johnson equation. The presentation is based on our results. In particular, we show that the Lax pairs corresponding to the KP and the CKP equations are gauge equivalent. We also describe some important classes of solutions obtained using the Darboux transformation approach. We present plots of exact solutions of the CKP equation including finite-gap solutions.


Johnson equation soliton finite-gap solution Darboux transformation lump 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Johnson, J. Fluid Mech., 97, 701–719 (1980).zbMATHCrossRefADSGoogle Scholar
  2. 2.
    V. D. Lipovskii, Izv. RAN Ser. Fiz. Atm. Okeana, 31, 664–871 (1995).Google Scholar
  3. 3.
    S. Leble and A. Sukhov, “On solutions of the Khokhlov-Zabolotskaya equation with KdV dispersion term,” in: Nonlinear Acoustics at the Turn of the Millennium (AIP Conf. Proc., Vol. 524, W. Lauterborn and T. Kurz, eds.), Amer. Inst. Phys., Melville, N. Y. (2000), p. 497–500.Google Scholar
  4. 4.
    L. A. Bordag, A. R. Its, S. V. Manakov, V. B. Matveev, and V. E. Zakharov, Phys. Lett. A, 63, 205–206 (1977).CrossRefADSGoogle Scholar
  5. 5.
    V. D. Lipovskii, V. B. Matveev, and A. O. Smirnov, J. Soviet Math., 46, 1609–1612 (1989).CrossRefGoogle Scholar
  6. 6.
    V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin (1991).zbMATHGoogle Scholar
  7. 7.
    B. B. Kadomtsev and V. I. Rydnik, Waves Around Us [in Russian], Znanie, Moscow (1981).Google Scholar
  8. 8.
    I. Anders and A. Boutet de Monvel, J. Nonlinear Math. Phys., 7, 284–302 (2000).zbMATHCrossRefGoogle Scholar
  9. 9.
    Yi-Tian Gao and Bo Tian, Phys. Lett. A, 301, 74–82 (2002).zbMATHCrossRefADSGoogle Scholar
  10. 10.
    V. S. Dryuma, Soviet Math. Dokl., 27, No. 1, 6–8 (1983).zbMATHGoogle Scholar
  11. 11.
    M. A. Salle, Candidate’s dissertation, LGU, Leningrad (1983).Google Scholar
  12. 12.
    W. Oevel and W.-H. Steeb, Phys. Lett. A, 103, 239–242 (1984).CrossRefADSGoogle Scholar
  13. 13.
    V. B. Matveev, Lett. Math. Phys., 3, 216–219 (1979).Google Scholar
  14. 14.
    E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin (1994).zbMATHGoogle Scholar
  15. 15.
    V. D. Lipovskii, Soviet Phys. Dokl., 31, No. 1, 31–33 (1986).ADSGoogle Scholar
  16. 16.
    J. Frauendiener and C. Klein, J. Comput. Appl. Math., 167, 193–218 (2004).zbMATHCrossRefADSGoogle Scholar
  17. 17.
    J. Frauendiener and C. Klein, Lett. Math. Phys., 76, 249–267 (2006).CrossRefGoogle Scholar
  18. 18.
    C. Klein and O. Richter, Ernst Equation and Riemann Surfaces: Analytical and Numerical Methods (Lect. Notes Phys., Vol. 685), Springer, Berlin (2005).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in SciencesLeipzigGermany
  2. 2.Institut de Mathématique de BourgogneDijonFrance
  3. 3.St. Petersburg University of Aerospace InstrumentationSt. PetersburgRussia

Personalised recommendations