Skip to main content
SpringerLink
Log in

Remark on the persistence of spatial analyticity for cubic nonlinear Schrödinger equation on the circle

  • Published:
Nonlinear Differential Equations and Applications NoDEA Aims and scope Submit manuscript

Abstract

We show that the uniform radius of spatial analyticity \(\sigma (t)\) of solutions at time t to the cubic nonlinear Schrödinger equations (NLS) on the circle cannot decay faster than 1 / t as \( t \rightarrow \infty \), given initial data that is analytic with fixed radius \(\sigma _0\). The same decay rate has been recently established for cubic NLS on the real line.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3(2), 107–156 (1993)

    Article  MathSciNet  Google Scholar 

  2. 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  Google Scholar 

  3. De Bouard, A., Hayashi, N., Kato, K.: Gevrey regularizing effect for the (generalized) Korteweg–de Vries equation and nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 12(6), 673–725 (1995)

    Article  MathSciNet  Google Scholar 

  4. Ferrari, A.B., Titi, E.S.: Gevrey regularity for nonlinear analytic parabolic equations. Commun. Partial Differ. Equ. 23(1–2), 1–16 (1998)

    MathSciNet  MATH  Google Scholar 

  5. Gérard, P., Guo, Y., Titi, E.S.: On the radius of analyticity of solutions to the cubic Szegő equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 32(1), 97–108 (2015)

    Article  MathSciNet  Google Scholar 

  6. Gorsky, J., Himonas, A.A.: Well-posedness of KdV with higher dispersion. Math. Comput. Simul. 80(1), 173–183 (2009)

    Article  MathSciNet  Google Scholar 

  7. Gorsky, J., Himonas, A.A., Holliman, C., Petronilho, G.: The Cauchy problem of a periodic higher order KdV equation in analytic Gevrey spaces. J. Math. Anal. Appl. 405(2), 349–361 (2013)

    Article  MathSciNet  Google Scholar 

  8. Hannah, H., Himonas, A.A., Petronilho, G.: Gevrey regularity of the periodic gKdV equation. J. Differ. Equ. 250(5), 2581–2600 (2011)

    Article  MathSciNet  Google Scholar 

  9. Hayashi, N.: Global existence of small analytic solutions to nonlinear Schrödinger equations. Duke Math. J. 60(3), 717–727 (1990)

    Article  MathSciNet  Google Scholar 

  10. Himonas, A.A., Henrik, K., Selberg, S.: On persistence of spatial analyticity for the dispersion-generalized periodic kdv equation. Nonlinear Anal. Real World Appl. 38, 35–48 (2017)

    Article  MathSciNet  Google Scholar 

  11. Himonas, A.A., Petronilho, G.: Analytic well-posedness of periodic gKdV. J. Differ. Equ. 253(11), 3101–3112 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Kato, T., Masuda, K.: Nonlinear evolution equations and analyticity. I. Ann. Inst. Henri Poincaré Anal. Non Linéaire 3(6), 455–467 (1986)

    Article  MathSciNet  Google Scholar 

  14. Katznelson, Y.: An Introduction to Harmonic Analysis, 3rd edn. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  15. Levermore, C.D., Oliver, M.: Analyticity of solutions for a generalized Euler equation. J. Differ. Equ. 133(2), 321–339 (1997)

    Article  MathSciNet  Google Scholar 

  16. Ma, Y.C., Ablowitz, M.J.: The periodic cubic Schrödinger equation. Stud. Appl. Math. 65(2), 113–158 (1981)

    Article  MathSciNet  Google Scholar 

  17. Oliver, M., Titi, E.S.: On the domain of analyticity of solutions of second order analytic nonlinear differential equations. J. Differ. Equ. 174(1), 55–74 (2001)

    Article  MathSciNet  Google Scholar 

  18. Panizzi, S.: On the domain of analyticity of solutions to semilinear Klein–Gordon equations. Nonlinear Anal. 75(5), 2841–2850 (2012)

    Article  MathSciNet  Google Scholar 

  19. Selberg, S., da Silva, D.O.: Lower bounds on the radius of spatial analyticity for the KdV equation. Ann. Henri Poincaré (2016). https://doi.org/10.1007/s00023-016-0498-1

    Article  MATH  Google Scholar 

  20. Selberg, S., Tesfahun, A.: On the radius of spatial analyticity for the 1d Dirac–Klein–Gordon equations. J. Differ. Equ. 259, 4732–4744 (2015)

    Article  MathSciNet  Google Scholar 

  21. Selberg, S., da Silva, D.O.: Lower bounds on the radius of spatial analyticity for the KdV equation. Ann. Henri Poincaré 18(3), 1009–1023 (2017)

    Article  MathSciNet  Google Scholar 

  22. Selberg, S., Tesfahun, A.: On the radius of spatial analyticity for the quartic generalized KdV equation. Ann. Henri Poincaré 18(11), 3553–3564 (2017)

    Article  MathSciNet  Google Scholar 

  23. Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol. 139. Springer, New York, Self-focusing and wave collapse (1999)

  24. Tao, T.: Nonlinear Dispersive Equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington; by the American Mathematical Society, Providence, Local and global analysis (2006)

  25. Tesfahun, A.: On the radius of spatial analyticity for cubic nonlinear Schrödinger equations. J. Differ. Equ. 263(11), 7496–7512 (2017)

    Article  MathSciNet  Google Scholar 

  26. Tesfahun, A.: Asymptotic lower bound for the radius of spatial analyticity to solutions of KdV equation. Commun. Contemp. Math. (2018). https://doi.org/10.1142/S021919971850061X

  27. Zygmund, A.: On Fourier coefficients and transforms of functions of two variables. Stud. Math. 50, 189–201 (1974)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bergen, PO Box 7803, 5020, Bergen, Norway

    Achenef Tesfahun

Authors
  1. Achenef Tesfahun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Achenef Tesfahun.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tesfahun, A. Remark on the persistence of spatial analyticity for cubic nonlinear Schrödinger equation on the circle. Nonlinear Differ. Equ. Appl. 26, 12 (2019). https://doi.org/10.1007/s00030-019-0558-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00030-019-0558-6

Keywords

Mathematics Subject Classification

Search

Navigation