Skip to main content
Log in

Long-Time Asymptotics for the Korteweg–de Vries Equation via Nonlinear Steepest Descent

  • Published:
Mathematical Physics, Analysis and Geometry Aims and scope Submit manuscript

Abstract

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg–de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ablowitz, M.J., Newell, A.C.: The decay of the continuous spectrum for solutions of the Korteweg–de Vries equation. J. Math. Phys. 14, 1277–1284 (1973)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Ablowitz, M.J., Segur, H.: Asymptotic solutions of the Korteweg–de Vries equation. Stud. Appl. Math. 57, 13–44 (1977)

    ADS  MathSciNet  Google Scholar 

  3. Beals, R., Coifman, R.: Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math. 37, 39–90 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beals, R., Deift, P., Tomei, C.: Direct and inverse scattering on the real line. In: Mathematical Surveys and Monographs, vol. 28. American Mathematical Society, Providence (1988)

  5. Budylin, A.M., Buslaev, V.S.: Quasiclassical integral equations and the asymptotic behavior of solutions of the Korteweg–de Vries equation for large time values. Dokl. Akad. Nauk 348(4), 455–458 (1996) (in Russian)

    MathSciNet  Google Scholar 

  6. Buslaev, V.S.: Use of the determinant representation of solutions of the Korteweg–de Vries equation for the investigation of their asymptotic behavior for large times. Uspekhi Mat. Nauk 36(4), 217–218 (1981) (in Russian)

    Google Scholar 

  7. Buslaev, V.S., Sukhanov, V.V.: Asymptotic behavior of solutions of the Korteweg–de Vries equation. J. Sov. Math. 34, 1905–1920 (1986)

    Article  MATH  Google Scholar 

  8. Deift, P.: Orthogonal polynomials and random matrices: a Riemann–Hilbert approach. In: Courant Lecture Notes, vol. 3. American Mathematical Society, Providence (1998)

    Google Scholar 

  9. Deift, P., Trubowitz, E.: Inverse scattering on the line. Comm. Pure Appl. Math. 32, 121–251 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  10. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Ann. Math. 137(2), 295–368 (1993)

    Article  MathSciNet  Google Scholar 

  11. Deift, P., Zhou, X.: Long time asymptotics for integrable systems. Higher order theory. Comm. Math. Phys. 165, 175–191 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Deift, P.A., Its, A.R., Zhou, X.: Long-time asymptotics for integrable nonlinear wave equations. In: Important Developments in Soliton Theory, Springer Ser. Nonlinear Dynam., pp. 181–204. Springer, Berlin (1993)

    Google Scholar 

  13. Deift, P., Venakides, S., Zhou, X.: The collisionless shock region for the long-time behavior of solutions of the KdV equation. Comm. Pure Appl. Math. 47, 199–206 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Deift, P., Kamvissis, S., Kriecherbauer, T., Zhou, X.: The Toda rarefaction problem. Comm. Pure Appl. Math. 49(1), 35–83 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Eckhaus, W., Schuur, P.: The emergence of solitons of the Korteweg–de Vries equation from arbitrary initial conditions. Math. Methods Appl. Sci. 5, 97–116 (1983)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Eckhaus, W., Van Harten, A.: The inverse scattering transformation and solitons: an introduction. In: Math. Studies, vol. 50. North-Holland, Amsterdam (1984)

    Google Scholar 

  17. Gardner, C.S., Green, J.M., Kruskal, M.D., Miura, R.M.: A method for solving the Korteweg–de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)

    Article  MATH  ADS  Google Scholar 

  18. Hirota, R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Article  ADS  Google Scholar 

  19. Its, A.R.: Asymptotic behavior of the solutions to the nonlinear Schrödinger equation, and isomonodromic deformations of systems of linear differential equations. Sov. Math., Dokl. 24(3), 452–456 (1981)

    MATH  Google Scholar 

  20. Its, A.R.: “Isomonodromy” solutions of equations of zero curvature. Math. USSR, Izv. 26(3), 497–529 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  21. Its, A.R.: Asymptotic behavior of the solution of the Cauchy problem for the modified Korteweg–de Vries equation . In: Wave Propagation. Scattering Theory, Probl. Mat. Fiz., vol. 12, pp. 214–224, 259. Leningrad. Univ., Leningrad (1987) (in Russian)

    Google Scholar 

  22. Its, A.R., Petrov, V.È.: “Isomonodromic” solutions of the sine-Gordon equation and the time asymptotics of its rapidly decreasing solutions. Sov. Math., Dokl. 26(1), 244–247 (1982)

    MATH  Google Scholar 

  23. Klaus, M.: Low-energy behaviour of the scattering matrix for the Schrödinger equation on the line. Inverse Problems 4, 505–512 (1988)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Krüger, H., Teschl, G.: Long-time asymptotics for the Toda lattice in the soliton region. Math. Z. 262, 585–602 (2009)

    Article  MATH  Google Scholar 

  25. Krüger, H., Teschl, G.: Long-time asymptotics of the Toda lattice for decaying initial data revisited. Rev. Math. Phys. 21(1), 61–109 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Manakov, S.V.: Nonlinear Frauenhofer diffraction. Sov. Phys. JETP 38(4), 693–696 (1974)

    ADS  MathSciNet  Google Scholar 

  27. Marchenko, V.A.: Sturm–Liouville Operators and Applications. Birkhäuser, Basel (1986)

    MATH  Google Scholar 

  28. McLaughlin, K.T.-R., Miller, P.D.: The \(\overline{\partial}\) steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights, (Art. ID 48673). IMRP Int. Math. Res. Pap. 2006, 1–77 (2006)

  29. Muskhelishvili, N.I.: Singular Integral Equations. P. Noordhoff, Groningen (1953)

    MATH  Google Scholar 

  30. Šabat, A.B.: On the Korteweg–de Vries equation. Sov. Math. Dokl. 14, 1266–1270 (1973)

    MATH  Google Scholar 

  31. Schuur, P.: Asymptotic analysis of soliton problems; an inverse scattering approach. In: Lecture Notes in Mathematics, vol. 1232. Springer, New York (1986)

    Google Scholar 

  32. Segur, H., Ablowitz, M.J.: Asymptotic solutions of nonlinear evolution equations and a Painléve transcendent. Phys. D 3, 165–184 (1981)

    Article  Google Scholar 

  33. Tanaka, S.: On the N-tuple wave solutions of the Korteweg–de Vries equation. Publ. Res. Inst. Math. Sci. 8, 419–427 (1972/73)

    Article  MathSciNet  Google Scholar 

  34. Tanaka, S.: Korteweg–de Vries equation; asymptotic behavior of solutions. Publ. Res. Inst. Math. Sci. 10, 367–379 (1975)

    Article  Google Scholar 

  35. Varzugin, G.G.: Asymptotics of oscillatory Riemann–Hilbert problems. J. Math. Phys. 37, 5869–5892 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. Wadati, M., Toda, M.: The exact N-soliton solution of the Korteweg–de Vries equation. Phys. Soc. Japan 32, 1403–1411 (1972)

    Article  ADS  Google Scholar 

  37. Zhou, X.: The Riemann–Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20(4), 966–986 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  38. Zakharov, V.E., Manakov, S.V.: Asymptotic behavior of nonlinear wave systems integrated by the inverse method. Sov. Phys. JETP 44, 106–112 (1976)

    ADS  MathSciNet  Google Scholar 

  39. Zabusky, N.J., Kruskal, M.D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gerald Teschl.

Additional information

Research supported by the Austrian Science Fund (FWF) under grant no. Y330.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grunert, K., Teschl, G. Long-Time Asymptotics for the Korteweg–de Vries Equation via Nonlinear Steepest Descent. Math Phys Anal Geom 12, 287–324 (2009). https://doi.org/10.1007/s11040-009-9062-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11040-009-9062-2

Keywords

Mathematics Subject Classifications (2000)

Navigation