Skip to main content
SpringerLink
Log in

Asymptotic behavior of solutions of the Korteweg-de Vries equation for large times

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

For the KdV equation a complete asymptotic expansion of the “dispersive tail” for large times is described, and generalized wave operators are introduced. The asymptotics for large times of the spectral Schrödinger equation with a potential of the type of a solution of the KdV equation is studied. It is shown that the KdV equation is connected in a specific manner with the structure of the asymptotics of solutions of the spectral equation. As a corollary, known explicit formulas for the leading terms of the asymptotics of solutions of the KdV equation in terms of spectral data corresponding to the initial conditions are obtained. A plan for justifying the results listed is outlined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature cited

  1. V. S. Buslaev and V. V. Sukhanov, “On the asymptotic behavior as t → ∞ of solutions of the equations ψXX + uψ + ψλ/4=0 with a potential satisfying the Korteweg-de Vries equation,” in: Probl. Mat. Fiz., Vol. 10, Leningrad State Univ. (1982), pp. 70–102.

  2. V. E. Zakharov and S. V. Manakov, “Asymptotic behavior of nonlinear wave systems integrable by the method of the inverse scattering problem,” Zh. Eksp. Teor. Fiz.,71, No. 1, 203–215 (1976).

    Google Scholar 

  3. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons. The Method of the Inverse Problem [in Russian], Moscow (1980).

  4. M. J. Ablowitz and H. Segur, “Asymptotic solutions of the Korteweg-de Vries equation,” Stud. Appl. Math.,57, 13–44 (1977).

    Google Scholar 

  5. C. S. Gardner, J. M. Green, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-de Vries equation,” Phys. Rev. Lett.,19, 1095–1097 (1967).

    Google Scholar 

  6. V. Yu. Novokshenov, “Asymptotics as t → ∞ of the solution of the Cauchy problem for the nonlinear Schrodinger equation,” Dokl. Akad. Nauk SSSR,251, No. 4, 799–802 (1980).

    Google Scholar 

  7. V. S. Buslaev, “Use of the determinant representation of solutions of the Kortewegde Vries equation for investigation of their asymptotic behavior for large times,” Usp. Mat. Nauk,36, No. 4, 217–218 (1981).

    Google Scholar 

  8. A. R. Its, “Asymptotics of solutions of the nonlinear Schrödinger equation and isomonodromic deformations of systems of linear differential equations,” Dokl. Akad. Nauk SSSR,261, No. 1, 14–18 (1981).

    Google Scholar 

  9. V. S. Buslaev and V. B. Matveev, “Wave operators for the Schrödinger equation with a slowly decaying potential,” Teor. Mat. Fiz.,2, No. 3, 367–376 (1970).

    Google Scholar 

Download references

Authors
  1. V. S. Buslaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Sukhanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 32–50, 1982.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Buslaev, V.S., Sukhanov, V.V. Asymptotic behavior of solutions of the Korteweg-de Vries equation for large times. J Math Sci 34, 1905–1920 (1986). https://doi.org/10.1007/BF01095099

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01095099

Keywords

Search

Navigation