SpringerLink
Log in

Lower bounds on the radius of spatial analyticity of solution for KdV-BBM type equations

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

Abstract

In this paper, we analyze the evolution of the radius of spatial analyticity to the solutions of generalized Korteweg-de Vries, Benjamin–Bona–Mahony (KdV-BBM) equation and coupled system of generalized Benjamin–Bona–Mahony (BBM) equations, subject to initial data which is analytic with a fixed radius \(\sigma _0\). It is shown that the uniform radius of spatial analyticity of solutions for both problems can not decay faster than \(ct^{-2/3}\) as \(t \rightarrow \infty \). We used the conservation law, contraction map** principle and different multilinear estimates to obtain the results.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Ahn, J., Kim, J., Seo, I.: On the radius of spatial analyticity for the Klein–Gordon–Schrödinger system. J. Differ. Equ. 321, 449–474 (2022)

    Article  MATH  Google Scholar 

  2. Ash, J.M., Cohen, J., Wang, G.: On strongly interacting internal solitary waves. J. Fourier Anal. Appl. 2(5), 507–517 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Belayneh, B., Tegegn, E., Tesfahun, A.: Lower bound on the radius of analyticity of solution for fifth order KdV-BBM equation. Nonlinear Differ. Equ. Appl. 29(1), 1–12 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bona, J., Tzvetkov, N.: Sharp well-posedness results for the BBM equation. Discrete Contin. Dyn. Syst. 23(4), 1241 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bona, J.L., Carvajal, X., Panthee, M., Scialom, M.: Higher-order Hamiltonian model for unidirectional water waves. J. Nonlinear Sci. 28(2), 543–577 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bona, J.L., Chen, H.: Local and global well-posedness results for generalized BBM-type equations. In: Evolution Equations, pp. 35–55. CRC Press (2019)

  7. Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory. J. Nonlinear Sci. 12(4)(2002)

  8. Bona, J.L., Grujić, Z.: Spatial analyticity properties of nonlinear waves. Math. Models Methods Appl. Sci. 13(03), 345–360 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bona, J.L., Grujić, Z., Kalisch, H.: Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation. Annales de l’IHP Analyse non linéaire 22, 783–797 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bona, J.L., Grujić, Z., Kalisch, H.: Global solutions of the derivative Schrödinger equation in a class of functions analytic in a strip. J. Differ. Equ. 229(1), 186–203 (2006)

    Article  MATH  Google Scholar 

  11. Chen, H., Haidau, C.A.: Well-posedness for the coupled BBM systems. J. Appl. Anal. Comput. 8(3), 890–914 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Cohen, J., Wang, G., et al.: Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities. Nagoya Math. J. 215, 67–149 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dufera, T.T., Mebrate, S., Tesfahun, A.: New lower bound for the radius of analyticity of solutions to the fifth order KdV-BBM model (2022). ar**v preprint ar**v:2203.08589

  14. Dufera, T.T., Mebrate, S., Tesfahun, A.: On the persistence of spatial analyticity for the beam equation. J. Math. Anal. Appl. 509(2), 126001 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  15. Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87(2), 359–369 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gear, J.A., Grimshaw, R.: Weak and strong interactions between internal solitary waves. Stud. Appl. Math. 70(3), 235–258 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Haidau, C.A.: A study of well posedness for systems of coupled non-linear dispersive wave equations. PhD thesis, University of Illinois at Chicago (2014)

  18. Himonas, A., Petronilho, G.: Evolution of the radius of spatial analyticity for the periodic BBM equation. Proc. Am. Math. Soc. 148(7), 2953–2967 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang, J., Wang, M.: New lower bounds on the radius of spatial analyticity for the KdV equation. J. Differ. Equ. 266(9), 5278–5317 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kato, T., Masuda, K.: Nonlinear evolution equations and analyticity. Annales de l’Institut Henri Poincare C Non Linear Anal. 3, 455–467 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge University Press (2004)

    Book  MATH  Google Scholar 

  22. Majda, A.J., Biello, J.A.: The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves. J. Atmos. Sci. 60(15), 1809–1821 (2003)

    Article  MathSciNet  Google Scholar 

  23. Mancas, S.C., Adams, R.: Elliptic solutions and solitary waves of a higher order KdV-BBM long wave equation. J. Math. Anal. Appl. 452(2), 1168–1181 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Petronilho, G., da Silva, P.L.: On the radius of spatial analyticity for the modified Kawahara equation on the line. Math. Nachr. 292(9), 2032–2047 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Selberg, S.: On the radius of spatial analyticity for solutions of the Dirac–Klein–Gordon equations in two space dimensions. Annales de l’Institut Henri Poincaré C Analyse non linéaire 36, 1311–1330 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Selberg, S., Tesfahun, A.: On the radius of spatial analyticity for the 1d Dirac–Klein–Gordon equations. J. Differ. Equ. 259(9), 4732–4744 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  30. Tesfahun, A.: Asymptotic lower bound for the radius of spatial analyticity to solutions of KdV equation. Commun. Contemp. Math. 21(08), 1850061 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tesfahun, A.: Remark on the persistence of spatial analyticity for cubic nonlinear Schrödinger equation on the circle. Nonlinear Differ. Equ. Appl. 26, 12 (2019)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

A. T acknowledges support from the Social Policy Research Grant (SPG), Nazarbayev University.

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Bahir Dar University, Bahir Dar, Ethiopia

    Emawayish Tegegn & Birilew Belayneh

  2. Department of Mathematics, Nazarbayev University, Qabanbai Batyr Avenue 53, 010000, Nur-Sultan, Republic of Kazakhstan

    Achenef Tesfahun

Authors
  1. Emawayish Tegegn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Achenef Tesfahun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Birilew Belayneh
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to this work.

Corresponding author

Correspondence to Emawayish Tegegn.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tegegn, E., Tesfahun, A. & Belayneh, B. Lower bounds on the radius of spatial analyticity of solution for KdV-BBM type equations. Nonlinear Differ. Equ. Appl. 30, 47 (2023). https://doi.org/10.1007/s00030-023-00855-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00030-023-00855-x

Keywords

Mathematics Subject Classification

Search

Navigation