Skip to main content
Log in

Asymptotic Expansions for One-Phase Soliton-Type Solutions of the Korteweg-De Vries Equation with Variable Coefficients

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We construct asymptotic expansions for a one-phase soliton-type solution of the Korteweg-de Vries equation with coefficients depending on a small parameter.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. D. J. Korteweg and G. de Vries, “On the change in form of long waves advancing in a rectangular canal and a new type of long stationary waves,” Phil. Mag., No. 39, 422–433 (1895).

    Google Scholar 

  2. J. Scott Russel, “Report on waves,” in: Rep. 14th Meeting of the British Association for the Advancement of Science, Murray, London (1844), pp. 311–390.

  3. N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solutions’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett., 15, 240–243 (1965).

    Article  Google Scholar 

  4. E. Fermi, J. Pasta, and S. Ulam, “Studies of nonlinear problems,” in: Collected Works of E. Fermi, Vol. II, Chicago University, Chicago (1965), pp. 978–988.

    Google Scholar 

  5. M. Toda, “Waves in nonlinear lattice,” Suppl. Theory Phys., No. 45, 174–200 (1970).

  6. G. M. Zaslavskii and R. Z. Sagdeev, Introduction to Nonlinear Physics. From Pendulum to Turbulence and Chaos [in Russian], Nauka, Moscow (1988).

    Google Scholar 

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

    Article  Google Scholar 

  8. P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Comm. Pure Appl. Math., 21, No.15, 467–490 (1968).

    Google Scholar 

  9. V. P. Maslov and M. V. Fedoryuk, Quasiclassical Approximation for Equations of Quantum Mechanics [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  10. A Discussion on Nonlinear Theory of Wave Propagation in Dispersive Systems [Russian translation], Mir, Moscow (1970).

  11. V. P. Maslov and G. A. Omel’yanov, “Asymptotic soliton-type solutions of equations with small dispersion,” Usp. Mat. Nauk, Issue 36 (219), No. 2, 63–124 (1981).

  12. V. A. Marchenko and E. Ya. Khruslov, Boundary-Value Problems in Domains with Fine-Grained Structure [in Russian], Naukova Dumka, Kiev (1974).

    Google Scholar 

  13. S. A. Lomov, Introduction to the General Theory of Singular Perturbations [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  14. G. B. Whitham, “Non-linear dispersive waves,” Proc. Roy. Soc., Ser. A, No. 283, 238–261 (1965).

  15. M. J. Lighthill, “A technique for rendering approximate solutions to physical problems uniformly valid, ” Phil. Mag., 40, 1179–1201 (1949).

    Google Scholar 

  16. S. Yu. Dobrokhotov and V. P. Maslov, “Finite-gap almost periodic solutions in the WKB approximations, ” in: VINITI Series in Contemporary Problems in Mathematics [in Russian], Issue 5, VINITI, Moscow (1980), pp. 3–94.

    Google Scholar 

  17. V. P. Maslov, Complex WKB Method in Nonlinear Equations [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  18. M. I. Vishik and L. A. Lyusternik, “Asymptotic behavior of solutions of linear differential equations with large or rapidly varying coefficients and boundary conditions,” Usp. Mat. Nauk, Issue 5 (121), 778–781 (1960).

  19. V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  20. V. G. Danilov and S. M. Frolovichev, “Tunnel WKB method for the construction of asymptotics of the Green function for parabolic equations,” Dokl. Ros. Akad. Nauk, 379, No.5, 591–594 (2001).

    Google Scholar 

  21. S. Yu. Dobrokhotov, I. N. Zhevandrov, V. P. Maslov, and A. N. Shafarevich, “Asymptotic rapidly decreasing solutions of linear strictly hyperbolic systems with variable coefficients,” Mat. Zametki, 49, No.4, 31–46 (1991).

    Google Scholar 

  22. R. Courant, Partial Differential Equations, Wiley, New York (1962).

    Google Scholar 

  23. V. K. Ivanov, “Associative algebra of the simplest generalized functions,” Sib. Mat. Zh., 20, No.4, 731–740 (1979).

    Google Scholar 

  24. G. A. Omel’yanov, “Interaction of waves of different scales in gas dynamics,” Mat. Zametki, 53, No.1, 148–151 (1993).

    Google Scholar 

  25. S. Yu. Dobrokhotov, “Hugoniot-Maslov chains for solitary vortices of the shallow water equations, ” Rus. J. Math. Phys., 6, No.2, 137–173 (1999).

    Google Scholar 

  26. R. Grimshaw, “Models for instability in inviscid fluid flows due to a resonance between two waves,” Nonlin. Instab. Anal., 2 1–14 (2001).

    Google Scholar 

  27. E. S. Benilov, R. Grimshaw, and E. P. Kuznetsova, “The generation of radiating waves in a singularly perturbed Korteweg-de Vries equation,” Physica D, 69, No.3-4, 270–278 (1993).

    Google Scholar 

  28. Yu. Samoilenko, “Asymptotical expansions for one-phase soliton-type solution to perturbed Korteweg-de Vries equation,” in: Proceedings of the Fifth International Conference “Symmetry in Nonlinear Mathematical Physics,” Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (2004).

    Google Scholar 

  29. V. V. Grushin, “On one class of elliptic pseudodifferential operators degenerating on a submanifold, ” Mat. Sb., Issue 84 (126), No. 2, 163–195 (1971).

Download references

Author information

Authors and Affiliations

Authors

Additional information

__________

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 111–124, January, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Samoilenko, V.H., Samoilenko, Y.I. Asymptotic Expansions for One-Phase Soliton-Type Solutions of the Korteweg-De Vries Equation with Variable Coefficients. Ukr Math J 57, 132–148 (2005). https://doi.org/10.1007/s11253-005-0176-9

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-005-0176-9

Keywords

Navigation