Skip to main content
SpringerLink
Log in

Stability of singular solutions to the b-family of equations

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we first construct some explicit solutions to the b-family of equations, which will become unbounded in a finite time. Then, we investigate the asymptotic stability of the aforementioned singular solutions of the b-family of equations in the Sobolev space \(H^s\) with \(s>\frac{7}{2}\). It is also interesting to point out that this stability highly depends on the values of parameter b, that is, \(b\in (-1,2]\). The proof is based on the detailed analysis on the estimates of the perturbed solutions and the properties of the corresponding linear operators.

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. Barnes, L., Hone, A.: Similarity reductions of peakon equation: the \(b\)-family. Theor. Math. Phys. 212, 1149–1167 (2022)

    Article  MathSciNet  Google Scholar 

  2. Bressan, A., Chen, G.: Lipschitz metric for a class of nonlinear wave equations. Arch. Ration. Mech. Anal. 226, 1303–1343 (2017)

    Article  MathSciNet  Google Scholar 

  3. Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)

    Article  MathSciNet  Google Scholar 

  4. Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 5, 1–27 (2007)

    Article  MathSciNet  Google Scholar 

  5. Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    Article  MathSciNet  Google Scholar 

  6. Cao, C., Holm, D., Titi, E.: Traveling wave solutions for a class of one-dimensional nonlinear shallow water wave models. J. Dyn. Differ. Equ. 16, 167–178 (2004)

    Article  MathSciNet  Google Scholar 

  7. Cai, H., Chen, G., Chen, R., Shen, Y.: Lipschitz metric for the Novikov equation. Arch. Ration. Mech. Anal. 229, 1091–1137 (2018)

    Article  MathSciNet  Google Scholar 

  8. Charalampidis, E., Parker, R., Kevrekidis, P., Lafortune, S.: The stability of the b-family of peakon equations. Nonlinearity 36, 1192–1217 (2023)

    Article  MathSciNet  Google Scholar 

  9. Chen, G., Chen, R., Liu, Y.: Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation. Indiana Univ. Math. J. 67, 2393–2433 (2018)

    Article  MathSciNet  Google Scholar 

  10. Coclite, G., Karlsen, K.: On the well-posedness of the Degasperis–Procesi equation. J. Funct. Anal. 233, 60–91 (2006)

    Article  MathSciNet  Google Scholar 

  11. Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)

    Article  MathSciNet  Google Scholar 

  12. Constantin, A., McKean, H.P.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52, 949–982 (1999)

    Article  MathSciNet  Google Scholar 

  13. Constantin, A., Strauss, W.: Stability of the Camassa–Holm solitons. J. Nonlinear Sci. 12, 415–422 (2002)

    Article  MathSciNet  Google Scholar 

  14. Constantin, A., Molinet, L.: Orbital stability of solitary waves for a shallow water equation. Physica D 157, 75–89 (2001)

    Article  MathSciNet  Google Scholar 

  15. Constantin, A., Strauss, W.: Stability of a class of solitary waves in compressible elastic rods. Phys. Lett. A 270, 140–148 (2000)

    Article  MathSciNet  Google Scholar 

  16. Degasperis, A., Holm, D., Hone, A.: A new integral equation with peakon solutions. Theor. Math. Phys. 133, 1463–1474 (2002)

    Article  Google Scholar 

  17. Degasperis, A., Holm, D., Hone, A.: Integrable and non-integrable equations with peakons. In: Ablowitz, M.J., Boiti, M., Pempinelli, F., Prinari, B. (eds.) Nonlinear Physics. Theory and Experiment II (Gallipoli, Italy, 27 June–6 July 2002), pp. 37–43. World Science, Singapore (2003). arXiv:nlin/0209008

  18. Degasperis, A., Procesi, M.: Asymptotic integrability. In: Symmetry and Perturbation Theory (Rome, 1998), pp. 23–37. World Scientific Publishing, River Edge (1999)

  19. Dullin, H., Gottwald, G., Holm, D.: Camassa–Holm, Korteweg–de Vries and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 33(1), 73–95 (2003)

    Article  MathSciNet  Google Scholar 

  20. Escher, J., Liu, Y., Yin, Z.: Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation. Indiana Univ. Math. J. 56, 87–117 (2007)

    Article  MathSciNet  Google Scholar 

  21. Escher, J., Yin, Z.: Well-posedness, blow-up phenomena, and global solutions for the b-equation. J. Reine Angew. Math. 624, 51–80 (2008)

    Article  MathSciNet  Google Scholar 

  22. Evans, L.: Partial Differential Equations, 2nd edn. American Mathematical Society (2010)

    Google Scholar 

  23. Gao, Y., Chen, J.: Stability of singular waves for Dullin–Gottwald–Holm equation. Nonlinear Anal. Real World Appl. 64, 103425 (2022)

    Article  MathSciNet  Google Scholar 

  24. Grayshan, K.: Continuity properties of the data-to-solution map for the periodic b-family equation. Differ. Integr. Equ. 25, 1–20 (2012)

    MathSciNet  Google Scholar 

  25. Guo, Z., Liu, X., Molinet, L., Yin, Z.: Ill-posedness of the Camassa–Holm and related equations in the critical space. J. Differ. Equ. 266, 1698–1707 (2019)

    Article  MathSciNet  Google Scholar 

  26. Hakkaev, S., Kirchev, K.: Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa–Holm equation. Commun. Partial Differ. Equ. 30, 761–781 (2005)

    Article  MathSciNet  Google Scholar 

  27. Himonas, A.A., Grayshan, K., Holliman, C.: Ill-Posedness for the b-Family of Equations. J. Nonlinear Sci. 26, 1175–1190 (2016)

    Article  MathSciNet  Google Scholar 

  28. Himonas, A., Holliman, C.: The Cauchy problem for a generalized Camassa–Holm equation. Adv. Differ. Equ. 19, 161–200 (2014)

    MathSciNet  Google Scholar 

  29. Holm, D., Staley, M.: Wave structures and nonlinear balances in a family of \(1+1\) evolutionary PDEs. Phys. Lett. A 308, 437–444 (2003)

    Article  MathSciNet  Google Scholar 

  30. Holm, D., Staley, M.: Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dyn. Syst. 3, 323–380 (2003)

    Article  MathSciNet  Google Scholar 

  31. Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations (Proceedings of Symposium), Dundee, 1974, Lecture Notes in Mathematics, Vol. 448, pp. 25–70. Springer, Berlin (1975)

  32. Lafortune, S., Pelinovsky, D.: Spectral instability of peakons in the b-family of the Camassa–Holm equations. SIAM J. Math. Anal. 54, 4572–4590 (2022)

    Article  MathSciNet  Google Scholar 

  33. Li, X., Huang, S., Yan, W.: Stability of blow-up solution for the two component Camassa–Holm equations. Asymptot. Anal. 120, 319–336 (2020)

    MathSciNet  Google Scholar 

  34. Li, H., Yan, W.: Asymptotic stability and instability of explicit self-similar waves for a class of nonlinear shallow water equations. Commun. Nonlinear Sci. Numer. Simul. 79, 104928 (2019)

    Article  MathSciNet  Google Scholar 

  35. Novruzov, E.: Construction of peakon-antipeakon solutions and ill-posedness for the b-family of equations. J. Differ. Equ. 272, 544–559 (2021)

    Article  MathSciNet  Google Scholar 

  36. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983)

    Book  Google Scholar 

  37. Zhang, S., Yin, Z.: Global solutions and blow-up phenomena for the periodic b-equation. J. Lond. Math. Soc. 82, 482–500 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referee very much for valuable comments and suggestions, in particular for pointing out Ref. [1] and references therein to their attention, which contribute to improve the manuscript. The work was supported by the Anhui Provincial Natural Science Foundation (2108085MA03) and research funds from Zhejiang Normal University (Nos. YS304222929 and ZZ323205020522016004).

Author information

Authors and Affiliations

  1. College of Mathematical Medicine, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Shou-Jun Huang

  2. Department of Mathematics, Anhui Normal University, Wuhu, 241002, People’s Republic of China

    Shou-Jun Huang & Li-Fan Wu

Authors
  1. Shou-Jun Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li-Fan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shou-Jun Huang.

Additional information

Communicated by Joachim Escher.

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

Huang, SJ., Wu, LF. Stability of singular solutions to the b-family of equations. Monatsh Math 204, 63–79 (2024). https://doi.org/10.1007/s00605-024-01964-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-024-01964-0

Keywords

Mathematics Subject Classification

Search

Navigation