Theoretical and Mathematical Physics

, Volume 180, Issue 1, pp 759–764 | Cite as

Five-wave classical scattering matrix and integrable equations

  • V. E. Zakharov
  • A. V. Odesskii
  • M. Cisternino
  • M. Onorato


We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.


integrability intermediate long-wave equation Korteweg-de Vries equation Benjamin-Ono equation scattering matrix 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. J. Ablowitz and H. Segur, Solitons, Nonlinear Evolution Equations, and Inverse Scattering, Cambridge Univ. Press, Cambridge (1991).CrossRefzbMATHGoogle Scholar
  2. 2.
    V. E. Zakharov, ed., What is Integrability?, Springer, Berlin (1991).Google Scholar
  3. 3.
    V. E. Zakharov and E. I. Schulman, Phys. D, 29, 283–320 (1988).CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    V. E. Zakharov and E. I. Schulman, Phys. D, 1, 191–202 (1980).CrossRefMathSciNetGoogle Scholar
  5. 5.
    V. E. Zakharov, A. Balk, and E. I. Schulman, “Conservation and scattering in nonlinear wave systems,” in: Important Developments in Soliton Theory (A. S. Fokas and V. E. Zakharov, eds.), Springer, New York (1993), pp. 375–404.CrossRefGoogle Scholar
  6. 6.
    A. I. Dyachenko, D. I. Kachulin, and V. E. Zakharov, JETP Lett., 98, 43–47 (2013).CrossRefADSGoogle Scholar
  7. 7.
    A. N. Hone and V. S. Novikov, J. Phys. A, 37, L399–L406 (2004).CrossRefMathSciNetADSGoogle Scholar
  8. 8.
    A. V. Mikhailov and V. S. Novikov, J. Phys. A, 35, 4775–4790 (2002); arXiv:nlin/0203055v1 (2002).CrossRefzbMATHMathSciNetADSGoogle Scholar
  9. 9.
    A. I. Dyachenko, Y. V. Lvov, and V. E. Zakharov, Phys. D, 87, 233–261 (1995).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    V. E. Zakharov, “Integrable systems in multidimensional spaces,” in: Mathematical Problems in Theoretical Physics (Lect. Notes Phys., Vol. 153, R. Schrader, R. Seiler, and D. A. Uhlenbrock, eds.), Springer, Berlin (1982), pp. 190–216.CrossRefGoogle Scholar
  11. 11.
    E. A. Zabolotskaya and R. V. Khokhlov, Sov. Phys. Acoustics, 15, 35–40 (1969).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • V. E. Zakharov
    • 1
    • 2
    • 3
  • A. V. Odesskii
    • 4
  • M. Cisternino
    • 5
  • M. Onorato
    • 5
    • 6
  1. 1.University of ArizonaTucsonUSA
  2. 2.Lebedev Physical InstituteRASMoscowRussia
  3. 3.Novosibirsk State UniversityNovosibirskRussia
  4. 4.Brock UniversitySt. CatharinesCanada
  5. 5.Dipartimento di FisicaUniverstà di TorinoTorinoItaly
  6. 6.INFNSezione di TorinoTorinoItaly

Personalised recommendations