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Wave breaking of the Camassa-Holm equation. (English) Zbl 1247.35104
Summary: This paper gives a new and direct proof for H. P. McKean’s theorem [Asian J. Math. 2, No. 4, 867–874 (1998; Zbl 0959.35140)] on wave breaking of the Camassa-Holm equation. The blow-up profile is also analyzed.
MSC:
35Q35PDEs in connection with fluid mechanics
35B44Blow-up (PDE)
37K40Soliton theory, asymptotic behavior of solutions
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
References:
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[2]Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 5(1), 1–27 (2007) · Zbl 1139.35378 · doi:10.1142/S0219530507000857
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[7]Fuchssteiner, B., Fokas, A.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D 4, 47–66 (1981/1982) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[8]Johnson, R.S.: Camassa–Holm, Korteweg–de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)
[9]Lenells, J.: Conservation laws of the Camassa–Holm equation. J. Phys. A 38(4), 869–880 (2005) · Zbl 1076.35100 · doi:10.1088/0305-4470/38/4/007
[10]McKean, H.: Breakdown of a shallow water equation. Asian J. Math. 2, 867–874 (1998)
[11]McKean, H.: Breakdown of the Camassa–Holm equation. Commun. Pure Appl. Math. 57, 416–418 (2004) · Zbl 1052.35130 · doi:10.1002/cpa.20003
[12]Whitham, G.B.: Linear and Nonlinear Waves. Wiley–Interscience, New York (1974)
[13]Zhou, Y.: Wave breaking for a shallow water equation. Nonlinear Anal. 57, 137–152 (2004) · Zbl 1106.35070 · doi:10.1016/j.na.2004.02.004