Breaking waves for the periodic two-component Camassa-Holm system

Guo, Fei

doi:10.1090/S0002-9939-2014-11949-6

Breaking waves for the periodic two-component Camassa-Holm system
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Proc. Amer. Math. Soc. 142 (2014), 2407-2415 Request permission

Abstract:

We give a blow-up result for the periodic two-component Camassa-Holm system modeling shallow water waves moving over a linear shear flow.

References

Roberto Camassa and Darryl D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), no. 11, 1661–1664. MR 1234453, DOI 10.1103/PhysRevLett.71.1661
R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation, Adv. in Appl. Mech. 31 (1994), 1-33.
C. X. Chen and S. Wen, Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system, submitted.
Adrian Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 321–362 (English, with English and French summaries). MR 1775353
A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci. 10 (2000), no. 3, 391–399. MR 1752603, DOI 10.1007/s003329910017
Adrian Constantin and Joachim Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998), no. 2, 229–243. MR 1668586, DOI 10.1007/BF02392586
Adrian Constantin and Joachim Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math. 51 (1998), no. 5, 475–504. MR 1604278, DOI 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
Adrian Constantin and Joachim Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z. 233 (2000), no. 1, 75–91. MR 1738352, DOI 10.1007/PL00004793
Adrian Constantin and Joachim Escher, On the structure of a family of quasilinear equations arising in shallow water theory, Math. Ann. 312 (1998), no. 3, 403–416. MR 1654808, DOI 10.1007/s002080050228
A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res. 40 (2008), no. 3, 175–211. MR 2369543, DOI 10.1016/j.fluiddyn.2007.06.004
Adrian Constantin and Rossen I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A 372 (2008), no. 48, 7129–7132. MR 2474608, DOI 10.1016/j.physleta.2008.10.050
Adrian Constantin and David Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), no. 1, 165–186. MR 2481064, DOI 10.1007/s00205-008-0128-2
H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech. 127 (1998), no. 1-4, 193–207. MR 1606738, DOI 10.1007/BF01170373
Joachim Escher, Olaf Lechtenfeld, and Zhaoyang Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst. 19 (2007), no. 3, 493–513. MR 2335761, DOI 10.3934/dcds.2007.19.493
Chunxia Guan and Zhaoyang Yin, Global weak solutions for a two-component Camassa-Holm shallow water system, J. Funct. Anal. 260 (2011), no. 4, 1132–1154. MR 2747018, DOI 10.1016/j.jfa.2010.11.015
Rossen Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion 46 (2009), no. 6, 389–396. MR 2598636, DOI 10.1016/j.wavemoti.2009.06.012
M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, Tsunami and nonlinear waves, Springer, Berlin, 2007, pp. 31–49. MR 2364923, DOI 10.1007/978-3-540-71256-5_{2}
Peter J. Olver and Philip Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E (3) 53 (1996), no. 2, 1900–1906. MR 1401317, DOI 10.1103/PhysRevE.53.1900
Zhaoyang Yin, On the blow-up of solutions of the periodic Camassa-Holm equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 12 (2005), no. 3-4, 375–381. MR 2127025
Yong Zhou, Blow-up of solutions to a nonlinear dispersive rod equation, Calc. Var. Partial Differential Equations 25 (2006), no. 1, 63–77. MR 2183855, DOI 10.1007/s00526-005-0358-1