Skip to main content
SpringerLink
Log in

Low regularity conservation laws for integrable PDE

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We present a general method for obtaining conservation laws for integrable PDE at negative regularity and exhibit its application to KdV, NLS, and mKdV. Our method works uniformly for these problems posed both on the line and on the circle.

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

References

  1. Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Studies in Appl. Math. 53(4), 249–315 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Apostol, T.: Mathematical analysis, 2nd edn. Addison-Wesley Publishing Co., Reading, MA (1974)

    MATH  Google Scholar 

  3. Bona, J.L.; Smith, R.: The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 278(1287), 555–601 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buckmaster, T.; Koch, H.: The Korteweg-de Vries equation at \(H^{-1}\) regularity. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(5), 1071–1098 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Carles and T. Kappeler. Norm-inflation for periodic NLS equations in negative Sobolev spaces. Preprint arXiv:1507.04218

  6. Christ, M.; Colliander, J.; Tao, T.: Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125(6), 1235–1293 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T.: Sharp global well-posedness for KdV and modified KdV on \({{\mathbb{R}}}\) and \({\mathbb{T}}\). J. Amer. Math. Soc. 16(3), 705–749 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T.: Multilinear estimates for periodic KdV equations, and applications. J. Funct. Anal. 211(1), 173–218 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fredholm, I.: Sur une classe d’équations fonctionnelles. Acta Math. 27(1), 365–390 (1903)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Z. Guo. Global well-posedness of Korteweg-de Vries equation in \(H^{-3/4}({{\mathbb{R} \it }})\). J. Math. Pures Appl. (9) 91(6) (2009), 583–597

  12. D. Hilbert. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (Erste Mitteilung). Nachr. Ges. Wiss. Göttingen (1904), 49–91

  13. Hirota, R.: Exact envelope-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14(7), 805–809 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Jost and A. Pais. On the scattering of a particle by a static potential. Physical Rev. (2) 82(6), (1951), 840–851

  15. T. Kappeler, C. Möhr, and P. Topalov. Birkhoff coordinates for KdV on phase spaces of distributions. Selecta Math. (N.S.) 11(1) (2005), 37–98

  16. Kappeler, T.; Topalov, P.: Global wellposedness of KdV in \(H^{-1}(\mathbb{T},\mathbb{R})\). Duke Math. J. 135(2), 327–360 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. T. Kato. On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Studies in applied mathematics, 93–128, Adv. Math. Suppl. Stud., 8, Academic Press, New York (1983).

  18. Kenig, C.E.; Ponce, G.; Vega, L.: A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9(2), 573–603 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Killip. On conservation laws for KdV. Research Seminar presented on December 9: at MSRI. CA, USA, Berkeley (2015)

    Google Scholar 

  20. Killip, R.; Conservation laws for integrable PDE. One hour lecture delivered March 16, : at the workshop “Singularity formation and long-time behavior in dispersive PDEs” held at the Mathematical Institute. University of Bonn, Germany (2016)

  21. Kishimoto, N.: Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity. Differential Integral Equations 22(5–6), 447–464 (2009)

    MathSciNet  MATH  Google Scholar 

  22. N. Kishimoto. A remark on norm inflation for nonlinear Schrödinger equations. Preprint

  23. H. Koch. Nonlinear dispersive equations. In: “Dispersive Equations and Nonlinear Waves”, 1–137, Oberwolfach Seminars, 45, Birkhauser/Springer Basel AG, Basel, (2014).

  24. H. Koch and D. Tataru. Conserved energies for the cubic NLS in 1-d. Preprint arXiv:1607.02534

  25. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467–490 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lax, P.D.: Periodic solutions of the KdV equation. Comm. Pure Appl. Math. 28, 141–188 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, B.: A priori bounds for KdV equation below \(H^{-3/4}\). J. Funct. Anal. 268(3), 501–554 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. R. M. Miura, C. S. Gardner, and M. D. Kruskal. Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Mathematical Phys. 9(8) (1968), 1204–1209

  29. Molinet, L.: A note on ill posedness for the KdV equation. Differential Integral Equations 24(7–8), 759–765 (2011)

    MathSciNet  MATH  Google Scholar 

  30. T. Oh. A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces. Preprint arXiv:1509.08143

  31. Sasa, N.; Satsuma, J.: New-type of soliton solutions for a higher-order nonlinear Schrödinger equation. J. Phys. Soc. Japan 60(2), 409–417 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. B. Simon. Trace ideals and their applications. Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI (2005).

  33. A. Sjöberg. On the Korteweg-de Vries equation. Report Dept of Computer Science Upsala University (1967).

  34. Sjöberg, A.: On the Korteweg-de Vries equation: existence and uniqueness. J. Math. Anal. Appl. 29, 569–579 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  35. R. Temam. Sur un problme non linéaire. J. Math. Pures Appl. (9) 48 (1969), 159–172

  36. Zakharov, V.E.; Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Ž. Èksper. Teoret. Fiz. 61(1), 118–134 (1971)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, CA, 90095, USA

    Rowan Killip & Monica Vişan

  2. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    Xiaoyi Zhang

Authors
  1. Rowan Killip
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Monica Vişan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaoyi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rowan Killip.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Killip, R., Vişan, M. & Zhang, X. Low regularity conservation laws for integrable PDE. Geom. Funct. Anal. 28, 1062–1090 (2018). https://doi.org/10.1007/s00039-018-0444-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-018-0444-0

Search

Navigation