Abstract
This paper gives a new and direct proof for McKean’s theorem (McKean in Asian J. Math. 2:867–874, 1998) on wave breaking of the Camassa–Holm equation. The blow-up profile is also analyzed.
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Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183(2), 215–239 (2007)
Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 5(1), 1–27 (2007)
Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)
Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier 50(2), 321–362 (2000)
Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192(1), 165–186 (2009)
Fokas, A.S., Liu, Q.M.: Asymptotic integrability of water waves. Phys. Rev. Lett. 77(12), 2347–2351 (1996)
Fuchssteiner, B., Fokas, A.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D 4, 47–66 (1981/1982)
Johnson, R.S.: Camassa–Holm, Korteweg–de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)
Lenells, J.: Conservation laws of the Camassa–Holm equation. J. Phys. A 38(4), 869–880 (2005)
McKean, H.: Breakdown of a shallow water equation. Asian J. Math. 2, 867–874 (1998)
McKean, H.: Breakdown of the Camassa–Holm equation. Commun. Pure Appl. Math. 57, 416–418 (2004)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley–Interscience, New York (1974)
Zhou, Y.: Wave breaking for a shallow water equation. Nonlinear Anal. 57, 137–152 (2004)
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Communicated by P. Newton.
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Jiang, Z., Ni, L. & Zhou, Y. Wave Breaking of the Camassa–Holm Equation. J Nonlinear Sci 22, 235–245 (2012). https://doi.org/10.1007/s00332-011-9115-0
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DOI: https://doi.org/10.1007/s00332-011-9115-0