SpringerLink
Log in

Orbital stability of periodic peakons for the higher-order modified Camassa–Holm equation

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

Considered herein is the orbital stability of the periodic peaked solitons for the higher-order modified Camassa–Holm equation. This equation can be viewed as a natural higher-order generalization of the modified Camassa–Holm equation, and admits a single peaked soliton and multi-peakons. We first show that the equation possesses the periodic peakons. Furthermore, it is proved that the periodic peakons are dynamically stable under small perturbations in the energy space by utilizing the inequalities with the maximum and minimum of the solutions related to the first two conservation laws.

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 includes VAT (Canada)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alber, M.S., Camassa, R., Holm, D.D., Marsden, J.E.: The geometry of peaked solitons and billiard solutions of a class of integrable PDEs. Lett. Math. Phys. 32, 137–151 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anco, S.C., Recio, E.: A general family of multi-peakon equations and their properties. J. Phys. A: Math. Theor. 52, 125203 (2019)

    Article  MATH  Google Scholar 

  3. Beals, R., Sattinger, D.H., Szmigielski, J.: Acoustic scattering and the extended Korteweg-de Vries hierarchy. Adv. Math. 140, 190–206 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Camassa, R., Holm, D.D., Hyman, J.M.: An new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, A.Y., Deng, T.J., Qiao, Z.J.: Stability of peakons and periodic peakons for a nonlinear quartic Camassa–Holm equation. Monatsh. Math. 198, 251–288 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, R.M., Lenells, J., Liu, Y.: Stability of the \(\mu \)-Camassa–Holm peakons. J. Nonlinear Sci. 23, 97–112 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50, 321–362 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Constantin, A.: On the scattering problem for the Camassa–Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457, 953–970 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Constantin, A., Strauss, W.A.: Stability of peakons. Comm. Pure Appl. Math. 53, 603–610 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ei Dika, K., Molinet, L.: Stability of multipeakons. Ann. Inst. H. Poincaré C Anal. Non Linéaire 26, 1517–1532 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ei Dika, K., Molinet, L.: Stability of multi Antipeakon-Peakons profile. Discrete Contin. Dyn. Syst. Ser. B 12, 561–577 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Fu, Y., Gui, G.L., Liu, Y., Qu, C.Z.: On the Cauchy problem for the integrable modified Camassa–Holm equation with cubic nonlinearity. J. Differ. Equ. 255, 1905–1938 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fuchssteiner, B.: Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa–Holm equation. Phys. D 95, 229–243 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their B\(\ddot{a}\)cklund transformations and hereditary symmetries. Phys. D 4, 47–66 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gui, G.L., Liu, Y., Olver, P.J., Qu, C.Z.: Wave-breaking and peakons for a modified Camassa–Holm equation. Comm. Math. Phys. 319, 731–759 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo, Z.H., Liu, X.C., Liu, X.X., Qu, C.Z.: Stability of peakons for the generalized modified Camassa–Holm equation. J. Differ. Equ. 266, 7749–7779 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Himonas, A.A., Mantzavinos, D.: The Cauchy problem for the Fokas–Olver-Rosenau–Qiao equation. Nonlinear Anal. 95, 499–529 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Johnson, R.S.: Camassa–Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lenells, J.: Stability of periodic peakons. Int. Math. Res. Not. 2004, 485–499 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lenells, J.: A variational approach to the stability of periodic peakons. J. Nonlinear Math. Phys. 11, 151–163 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, Y.A., Olver, P.J.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Equ. 162, 27–63 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, X.X.: Orbital stability of peakons for a modified Camassa–Holm equation with higher-order nonlinearity. Discrete Contin. Dyn. Syst. 38, 5505–5521 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, X.X.: Stability in the energy space of the sum of \(N\) peakons for a modified Camassa–Holm equation with higher-order nonlinearity. J. Math. Phys. 59, 121505 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, X.C., Liu, Y., Olver, P.J., Qu, C.Z.: Orbital stability of peakons for a generalization of the modified Camassa–Holm equation. Nonlinearity 27, 2297–2319 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, Y., Qu, C.Z., Zhang, Y.: Stability of periodic peakons for the modified \(\mu \)-Camassa–Holm equation. Phys. D 250, 66–74 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Martel, Y., Merle, F., Tsai, T.P.: Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations. Comm. Math. Phys. 231, 347–373 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Moon, B.: Orbital stability of periodic peakons for the generalized modified Camassa–Holm equation. Discrete Contin. Dyn. Syst. Ser. S 14, 4409–4437 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  33. Olver, P.J., Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E 53, 1900–1906 (1996)

    Article  MathSciNet  Google Scholar 

  34. Qiao, Z.J.: A new integrable equation with cuspons and W/M-shape-peaks solitons. J. Math. Phys. 47, 112701 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Qiao, Z.J., Li, X.Q.: An integrable equation with nonsmooth solitons. Theoret. and Math. Phys. 167, 584–589 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Qu, C.Z., Fu, Y., Liu, Y.: Well-posedness, wave breaking and peakons for a modified \(\mu \)-Camassa–Holm equation. J. Funct. Anal. 266, 433–477 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Qu, C.Z., Liu, X.C., Liu, Y.: Stability of peakons for an integrable modified Camassa–Holm equation with cubic nonlinearity. Comm. Math. Phys. 322, 967–997 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Qu, C.Z., Zhang, Y., Liu, X.C., Liu, Y.: Orbital stability of periodic peakons to a generalized \(\mu \)-Camassa–Holm equation. Arch. Ration. Mech. Anal. 211, 593–617 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. **a, B.Q., Qiao, Z.J., Li, J.B.: An integrable system with peakon, complex peakon, weak kink, and kink-peakon interactional solutions. Commun. Nonlinear Sci. Numer. Simul. 63, 292–306 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yang, M.L., Li, Y.S., Zhao, Y.Y.: On the Cauchy problem of generalized Fokas–Olver-Rosenau–Qiao equation. Appl. Anal. 97, 2246–2268 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yin, J.L., Tian, L.X.: Stability of peakons and linear dispersion limit for the periodic Dullin–Gottwald–Holm equation. J. Math. Phys. 51, 023505 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work of Chong is supported by the National NSF of China Grant-11631007. The work of Fu is supported by the National NSF of China Grants-11471259 and 11631007 and Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 22JSY003).

Author information

Authors and Affiliations

  1. Center for Nonlinear Studies and School of Mathematics, Northwest University, **’an, 710127, People’s Republic of China

    Gezi Chong & Ying Fu

  2. School of Science, **’an University of Architecture and Technology, **’an, 710055, People’s Republic of China

    Hao Wang

Authors
  1. Gezi Chong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ying Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ying Fu.

Additional information

Communicated by Adrian Constantin.

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

Chong, G., Fu, Y. & Wang, H. Orbital stability of periodic peakons for the higher-order modified Camassa–Holm equation. Monatsh Math (2023). https://doi.org/10.1007/s00605-023-01906-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00605-023-01906-2

Keywords

Mathematics Subject Classification

Search

Navigation