Skip to main content
SpringerLink
Log in

Blow-Up Solutions and Peakons to a Generalized μ-Camassa–Holm Integrable Equation

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Considered here is a generalized μ-type integrable equation, which can be regarded as a generalization to both the μ-Camassa–Holm and modified μ-Camassa–Holm equations. It is shown that the proposed equation is formally integrable with the Lax-pair and the bi-Hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary-gravity waves. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established by the viscosity method. Existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are investigated. It is found that the equation admits single and multi-peaked traveling wave solutions. The effects of varying μ-Camassa–Holm and modified μ-Camassa–Holm nonlocal nonlinearities on blow-up criteria and wave breaking are illustrated in detail. Our analysis relies on the method of characteristics and conserved quantities and is proceeded with a priori differential estimates.

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations. Grundlehren der mathematischen Wissenschaften 343. Springer, Berlin (2011)

  3. Bona J.L., Smith R.: The initial value problem for the Korteweg-de Vries equation. Philos. Trans. R. Soc. Lond. A. 278, 555–601 (1975)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Chou K.S., Qu C.Z.: Integrable equations arising from motions of plane curves I. Phys. D. 162, 9–33 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  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  MATH  MathSciNet  Google Scholar 

  10. Constantin A.: On the blow-up of solutions of a periodic shallow water equation. J. Nonlinear Sci. 10, 391–399 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Constantin A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Constantin A., Escher J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 233, 75–91 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Constantin A., Escher J.: Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa 26, 303–328 (1998)

    MATH  MathSciNet  Google Scholar 

  15. Constantin A., Escher J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Constantin A., Gerdjikov V.S., Ivanov R.I.: Inverse scattering transform for the Camassa–Holm equation. Inverse Probl. 22, 2197–2207 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Constantin A., Gerdjikov V., Ivanov R.: Generalised Fourier transform for the Camassa–Holm hierarchy. Inverse Probl. 23, 1565–1597 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Constantin A., Kappeler T., Kolev B., Topalov P.: On geodesic exponential maps of the Virasoro group. Ann. Glob. Anal. Geom. 31, 155–180 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Danchin R.: A few rearks on the Camassa–Holm equation. Differ. Integral Equ. 14, 953–988 (2001)

    MATH  MathSciNet  Google Scholar 

  22. El Dika K., Molinet L.: Stability of multipeakons. Ann. Inst. H. Poincaré Anal. Non Lin’eaire 18, 1517–1532 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  23. Faquir M., Manna M.A., Neveu A.: An integrable equation governing short waves in a long-wave model. Proc. R. Soc. A 463, 1939–1954 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Fokas A.S.: On a class of physically important integrable equations. Phys. D. 87, 145–150 (2002)

    Article  MathSciNet  Google Scholar 

  25. Fu Y., Liu Y., Qu C.Z.: On the blow-up structure for the generalized periodic Camassa–Holm and Degasperis–Procesi equations. J. Funct. Anal. 262, 3125–3158 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Fuchssteiner B., Fokas A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D. 4, 47–66 (1981/1982)

    Article  MathSciNet  Google Scholar 

  28. Gui G., Liu Y.: On the global existence and wave-breaking criteria for the two-component Camassa–Holm system. J. Funct. Anal. 258, 4251–4278 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

  32. Khesin B., Lenells J., Misiołek G.: Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms. Math. Ann. 342, 617–656 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. Kouranbaeva S.: The Camassa–Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40, 857–868 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  34. Lenells J.: The scattering approach for the Camassa–Holm equation. J. Nonlinear Math. Phys. 9, 389–393 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. Lenells J., Misiolek G., Tiğlay F.: Integrable evolution equations on spaces of tensor densities and their peakon solutions. Commun. Math. Phys. 299, 129–161 (2010)

    Article  ADS  MATH  Google Scholar 

  36. 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  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  38. Misiołek G.: A shallow water equation as a geodesic flow on the Bott–Virasoro group. J. Geom. Phys. 24, 203–208 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Olver P.J.: Invariant submanifold flows. J. Phys. A. 4, 344017 (2008)

    Article  MathSciNet  Google Scholar 

  40. 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  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  42. Qiao, Z.J., Xia, B.Q., Li, J.B.: Integrable systems with peakons, weak kinks, anti-kink peakon interactional solutions (2012) (preprint)

  43. Qu C.Z., Fu Y., Liu Y.: Well-posedness, wave breaking and peakons for a modified μ-Camassa–Holm equation. J. Funct. Anal. 266(2), 422–477 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  45. Schäfer T., Wayne C.E.: Propagation of ultra-short optical pulses in cubic nonlinear media. Phys. D. 196, 90–105 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  46. Schiff J.: The Camassa–Holm equation: a loop group approach. Phys. D 121, 24–43 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  47. Toland J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Changzheng Qu & Yue Liu

  2. Department of Mathematics, Northwest University, Xi’an, 710069, People’s Republic of China

    Ying Fu

  3. Department of Mathematics, University of Texas, Arlington, TX, 76019-0408, USA

    Yue Liu

Authors
  1. Changzheng Qu
    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. Yue Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yue Liu.

Additional information

Communicated by H.-T. Yau

Rights and permissions

Reprints and permissions

About this article

Cite this article

Qu, C., Fu, Y. & Liu, Y. Blow-Up Solutions and Peakons to a Generalized μ-Camassa–Holm Integrable Equation. Commun. Math. Phys. 331, 375–416 (2014). https://doi.org/10.1007/s00220-014-2007-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2007-z

Keywords

Search

Navigation