Skip to main content
SpringerLink
Log in

New lower bounds for the radius of analyticity for the mKdV equation and a system of mKdV-type equations

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

This paper is devoted to obtaining new lower bounds to the radius of spatial analyticity for the solutions of modified Korteweg–de Vries (mKdV) equation and a coupled system of mKdV-type equations, starting with real analytic initial data with a fixed radius of analyticity \(\sigma _0\). Specifically, we derive almost conserved quantities to prove that the local solution can be extended to a time interval [0, T] for any large \(T>0\) in such a way that the radius of analyticity \(\sigma (T)\) decays no faster than \(cT^{-1}\) for both the equations, where c is a positive constant. The results of this paper improve the ones obtained in Figueira and Panthee (Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion, to appear in NoDEA, arXiv:2307.09096) and Figueira and Himonas (J Math Anal Appl 497(2):124917, 2021), respectively, for the mKdV equation and a mKdV-type system.

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

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. M. Ablowitz, D. Kaup, A. Newell and H. Segur, Nonlinear evolution equations of physical significance, Phys. Rev. Lett., 31(2) (1973) 125–127.

    Article  MathSciNet  Google Scholar 

  2. E. Alarcon, J. Angulo and J. F. Montenegro, Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Analysis, 36 (1999) 1015–1035.

  3. R. Barostichi, R. O. Figueira and A. A. Himonas, The modified KdV equation with higher dispersion in Sobolev and analytic spaces on the line, J. Evol. Equ. 21 (2021), 2213-2237.

    Article  MathSciNet  Google Scholar 

  4. B. Belayneh, E. Tegegn and A. Tesfahun, Lower bound on the radius of analyticity of solution for fifth order KdV-BBM equation, Nonlinear Differ. Equ. Appl. (2022) 29:6

    Article  MathSciNet  Google Scholar 

  5. J. L. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann Inst. H. Poincaré 22 (2005) 783–797.

    MathSciNet  Google Scholar 

  6. X. Carvajal and M. Panthee, Sharp well-posedness for a coupled system of mKdV-type equations. J. Evol. Equ. 19 (4), (2019), 1167–1197.

    Article  MathSciNet  Google Scholar 

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

  8. A. Corcho, M. Panthee, Global well-posedness for a coupled modified KdV system, Bull Braz Math Soc, New Series 43, (2012) 27–57.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. L. G. Farah, Global rough solutions to the critical generalized KdV equation, J. Differential Equations 249 (2010) 1968–1985.

    Article  MathSciNet  Google Scholar 

  11. R. O. Figueira and M. Panthee, Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion, (to appear in Nonlinear Differential Equations and Applications, NoDEA), arXiv:2307.09096

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

  13. R. O. Figueira and A. A. Himonas, Lower bounds on the radius of analyticity for a system of modified KdV equations, J. Math. Anal. Appl. 497, No. 2, (2021) 124917.

    Article  MathSciNet  Google Scholar 

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

  15. Z. Grujić, H. Kalisch; Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations, 15 (2002) 1325–1334.

    Article  MathSciNet  Google Scholar 

  16. A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential and Integral Equations, 18 (2005) 133-1339.

    Article  MathSciNet  Google Scholar 

  17. Huang, J., Wang, M. New lower bounds on the radius of spatial analyticity for the KdV equation, J. Differ. Equations 266, No. 9, (2019), 5278–5317.

  18. T. Kato, K. Masuda, Nonlinear evolution equations and analyticity I, Ann. Inst. Henri Poincaré , 3 (1986) 455-467.

    Article  MathSciNet  Google Scholar 

  19. C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993) 527–620.

    Article  MathSciNet  Google Scholar 

  20. C. E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991) 33–69.

    Article  MathSciNet  Google Scholar 

  21. R. M. Miura, The Korteweg-de Vries Equation: A Survey of Results, SIAM Review 18 No. 3 (1976) 412–459.

    Article  MathSciNet  Google Scholar 

  22. S. Selberg and D. O. Silva, Lower bounds on the radius of a spatial Analyticity for the KdV equation, Ann. Henri Poincarè, 18 (2017) 1009–1023.

    Article  MathSciNet  Google Scholar 

  23. S. Selberg and A. Tesfahun, On the radius of spatial analyticity for the 1d Dirac-Klein-Gordon equations, Journal of Differential Equations 259 (2015), 4732–4744.

    Article  MathSciNet  Google Scholar 

  24. T. Tao, Multilinear weighted convolution of \(L^2\) functions and applications to nonlinear dispersive equations, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the Amer. J. math, 123 (2001), 839–908.

  25. E. Tegegn, A. Tesfahun and B. Belayneh, Lower bounds on the radius of spatial analyticity of solution for KdV-BBM type equations, Nonlinear Differ. Equ. Appl. (2023) 30:47.

    Article  MathSciNet  Google Scholar 

  26. A. Tesfahun, On the radius of spatial analyticity for cubic nonlinear Schrödinger equations, J. of Differential Equations, 263, No. 11, (2017) 7496–7512.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author acknowledges the support from FAPESP, Brazil (#2021/04999-9). The second author acknowledges the grants from FAPESP, Brazil (#2023/06416-6) and CNPq, Brazil (#307790/2020-7). The authors would also like to thank the unanimous referees whose comments helped immensely to improve the original manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Campinas, Campinas, SP, 13083-859, Brazil

    Renata O. Figueira & Mahendra Panthee

Authors
  1. Renata O. Figueira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mahendra Panthee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mahendra Panthee.

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

Figueira, R.O., Panthee, M. New lower bounds for the radius of analyticity for the mKdV equation and a system of mKdV-type equations. J. Evol. Equ. 24, 42 (2024). https://doi.org/10.1007/s00028-024-00977-4

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00028-024-00977-4

Keywords

Mathematics Subject Classification

Search

Navigation