Mathematical Physics, Analysis and Geometry

, Volume 12, Issue 3, pp 287–324 | Cite as

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

Article

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.

Keywords

Riemann–Hilbert problem KdV equation Solitons 

Mathematics Subject Classifications (2000)

Primary 37K40 35Q53 Secondary 37K45 35Q15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Ablowitz, M.J., Segur, H.: Asymptotic solutions of the Korteweg–de Vries equation. Stud. Appl. Math. 57, 13–44 (1977)ADSMathSciNetGoogle Scholar
  3. 3.
    Beals, R., Coifman, R.: Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math. 37, 39–90 (1984)MATHCrossRefMathSciNetGoogle Scholar
  4. 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)Google Scholar
  5. 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)MathSciNetGoogle Scholar
  6. 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. 7.
    Buslaev, V.S., Sukhanov, V.V.: Asymptotic behavior of solutions of the Korteweg–de Vries equation. J. Sov. Math. 34, 1905–1920 (1986)MATHCrossRefGoogle Scholar
  8. 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. 9.
    Deift, P., Trubowitz, E.: Inverse scattering on the line. Comm. Pure Appl. Math. 32, 121–251 (1979)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Ann. Math. 137(2), 295–368 (1993)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Deift, P., Zhou, X.: Long time asymptotics for integrable systems. Higher order theory. Comm. Math. Phys. 165, 175–191 (1994)MATHCrossRefADSMathSciNetGoogle Scholar
  12. 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. 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)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Deift, P., Kamvissis, S., Kriecherbauer, T., Zhou, X.: The Toda rarefaction problem. Comm. Pure Appl. Math. 49(1), 35–83 (1996)MATHCrossRefMathSciNetGoogle Scholar
  15. 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)MATHCrossRefADSMathSciNetGoogle Scholar
  16. 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. 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)MATHCrossRefADSGoogle Scholar
  18. 18.
    Hirota, R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)CrossRefADSGoogle Scholar
  19. 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)MATHGoogle Scholar
  20. 20.
    Its, A.R.: “Isomonodromy” solutions of equations of zero curvature. Math. USSR, Izv. 26(3), 497–529 (1986)MATHCrossRefMathSciNetGoogle Scholar
  21. 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. 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)MATHGoogle Scholar
  23. 23.
    Klaus, M.: Low-energy behaviour of the scattering matrix for the Schrödinger equation on the line. Inverse Problems 4, 505–512 (1988)MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Krüger, H., Teschl, G.: Long-time asymptotics for the Toda lattice in the soliton region. Math. Z. 262, 585–602 (2009)MATHCrossRefGoogle Scholar
  25. 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)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Manakov, S.V.: Nonlinear Frauenhofer diffraction. Sov. Phys. JETP 38(4), 693–696 (1974)ADSMathSciNetGoogle Scholar
  27. 27.
    Marchenko, V.A.: Sturm–Liouville Operators and Applications. Birkhäuser, Basel (1986)MATHGoogle Scholar
  28. 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)Google Scholar
  29. 29.
    Muskhelishvili, N.I.: Singular Integral Equations. P. Noordhoff, Groningen (1953)MATHGoogle Scholar
  30. 30.
    Šabat, A.B.: On the Korteweg–de Vries equation. Sov. Math. Dokl. 14, 1266–1270 (1973)MATHGoogle Scholar
  31. 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. 32.
    Segur, H., Ablowitz, M.J.: Asymptotic solutions of nonlinear evolution equations and a Painléve transcendent. Phys. D 3, 165–184 (1981)CrossRefGoogle Scholar
  33. 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)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Tanaka, S.: Korteweg–de Vries equation; asymptotic behavior of solutions. Publ. Res. Inst. Math. Sci. 10, 367–379 (1975)CrossRefGoogle Scholar
  35. 35.
    Varzugin, G.G.: Asymptotics of oscillatory Riemann–Hilbert problems. J. Math. Phys. 37, 5869–5892 (1996)MATHCrossRefADSMathSciNetGoogle Scholar
  36. 36.
    Wadati, M., Toda, M.: The exact N-soliton solution of the Korteweg–de Vries equation. Phys. Soc. Japan 32, 1403–1411 (1972)CrossRefADSGoogle Scholar
  37. 37.
    Zhou, X.: The Riemann–Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20(4), 966–986 (1989)MATHCrossRefMathSciNetGoogle Scholar
  38. 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)ADSMathSciNetGoogle Scholar
  39. 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)CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaWienAustria
  2. 2.International Erwin Schrödinger Institute for Mathematical PhysicsWienAustria

Personalised recommendations