Publication Cover
Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 17
192
Views
1
CrossRef citations to date
0
Altmetric
Research Article

Orbital stability of periodic peakons for a generalized Camassa–Holm equation

Pages 6136-6150 | Received 29 Mar 2020, Accepted 01 Aug 2020, Published online: 05 May 2021

References

  • Hakkaev S, Kirchev K. Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation. Commun Part Diff Equ. 2005;30:761–781.
  • Camassa R, Holm DD. An integrable shallow water equation with peaked solitons. Phys Rev Lett. 1993;71:1661–1664.
  • Fuchssteiner B, Fokas AS. Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D. 1981/1982;4:47–66.
  • Constantin A, Lannes D. The hydrodynamical relevance of the Camassa- Holm and Degasperis-Procesi equations. Arch Ration Mech Anal. 2009;192:165–186.
  • Johnson RS. Camassa-Holm, Korteweg-de Vries and related models for water waves. J Fluid Mech. 2002;455:63–82.
  • Fuchssteiner B. Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation. Physica D. 1996;95:229–243.
  • Olver PJ, Rosenau P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys Rev E. 1996;53:1900–1906.
  • Chou KS, Qu CZ. Integrable equations arising from motions of plane curves I. Physica D. 2002;162:9–33.
  • Constantin A, Kappeler T, Kolev B, et al. On geodesic exponential maps of the Virasoro group. Ann Glob Anal Geom. 2007;31:155–180.
  • Kouranbaeva S. The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. J Math Phys. 1999;40:857–868.
  • Misiolek G. A shallow water equation as a geodesic flow on the Bott-Virasoro group. J Geom Phys. 1998;24:203–208.
  • Constantin A. Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann Inst Fourier (Grenoble). 2000;50:321–362.
  • Constantin A, Escher J. Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 1998;181:229–243.
  • Constantin A, Escher J. On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math Z. 2000;233:75–91.
  • Constantin A, Escher J. Global existence and blow-up for a shallow water equation. Ann Scuola Norm Sup Pisa. 1998;26:303–328.
  • Li YA, Olver P. Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J Differ Equ. 2000;162:27–63.
  • Anco SC, Recio E, Gandarias ML, et al. A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Dynamical systems, differential equations and applications. Proceedings of the 10th AIMS International Conference; Madrid, Spain: 2015. p. 29–37.
  • Constantin A, Strauss W. Stability of peakons. Comm Pure Appl Math. 2000;53:603–610.
  • Hakkaev S, Kirchev K. On the well-posedness and stability of peakons for a generalized Camassa-Holm equation. Int J Nonlinear Sci. 2006;1:139–148.
  • Lin ZW, Liu Y. Stability of peakons for the Degasperis-Procesi equation. Comm Pure Appl Math. 2009;62:125–146.
  • Qu CZ, Liu XC, Liu Y. Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity. Comm Math Phys. 2013;322:967–997.
  • Liu XC, Yue L, Olver PJ, et al. Orbital stability of Peakons for a generalization of the modified Camassa-Holm equation. Nonlinearity. 2014;27:2297–2319.
  • Constantin A, Molinet L. Orbital stability of solitary waves for a shallow water equation. Physica D. 2001;157:75–89.
  • Chen RM, Liu XC, Liu Y, et al. Stability of the Camassa-Holm peakons in the dynamics of a shallow-water-type system. Calc Var Partial Differ Equ. 2016;55:34.
  • Lenells J. Stability of periodic peakons. Int Math Res Not. 2004;10:485–499.
  • Chen M, Lenells J, Liu Y. Stability of the μ-Camassa-Holm peakons. J Nonlinear Sci. 2013;23:97–112.
  • Liu Y, Qu CZ, Zhang Y. Stability of peakons for the modified μ-Camassa-Holm equation. Physica D. 2013;250:66–74.
  • Qu CZ, Zhang Y, Liu XC, et al. Orbital stability of periodic peakons to a generalized mu-Camassa-Holm equation. Arch Rational Mech Anal. 2014;211:593–617.
  • Kato T. Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral theory and differential equations. Everitt WN, editors, Berlin: Springer Verlag; 1995. p. 25–70. (Lecture Notes in Math.; 448).

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.