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Global conservative and multipeakon conservative solutions for the modified Camassa-Holm system with coupling effects. (English) Zbl 1407.35181
Summary: This paper investigates the continuation of solutions to the modified coupled two-component Camassa-Holm system after wave breaking. The underlying problem is rather challenging due to the mutual coupling effect between two components in the system. By introducing a novel transformation that makes use of a skillfully defined characteristic and a set of newly defined variables, the original system is converted into a Lagrangian equivalent system, from which the global conservative solution is obtained, which further allows for the establishment of the multipeakon conservative solution of the system. The results obtained herein are deemed useful for understanding the inevitable phenomenon near wave breaking.
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B60 Continuation and prolongation of solutions to PDEs
35M31 Initial value problems for mixed-type systems of PDEs
Full Text: DOI
[1] Fu, Y.; Liu, Y.; Qu, C., Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons, Mathematische Annalen, 348, 2, 415-448, (2010) · Zbl 1207.35074
[2] Fu, Y.; Qu, C., Well posedness and blow-up solution for a new coupled Camassa-Holm equations with peakons, Journal of Mathematical Physics, 50, 1, article 012906, (2009) · Zbl 1189.35273
[3] Constantin, A., The Hamiltonian structure of the Camassa-Holm equation, Expositiones Mathematicae, 15, 1, 53-85, (1997)
[4] Camassa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons, Physical Review Letters, 71, 11, 1661-1664, (1993) · Zbl 0972.35521
[5] Constantin, A., On the scattering problem for the Camassa-Holm equation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 457, 2008, 953-970, (2001) · Zbl 0999.35065
[6] Dullin, H. R.; Gottwald, G. A.; Holm, D. D., An integrable shallow water equation with linear and nonlinear dispersion, Physical Review Letters, 87, 19, (2001)
[7] Constantin, A.; Lannes, D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Archive for Rational Mechanics and Analysis, 192, 1, 165-186, (2009) · Zbl 1169.76010
[8] Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181, 2, 229-243, (1998) · Zbl 0923.76025
[9] Constantin, A., Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Annales de l’Institut Fourier, 50, 2, 321-362, (2000) · Zbl 0944.00010
[10] Bressan, A.; Constantin, A., Global conservative solutions of the Camassa-Holm equation, Archive for Rational Mechanics and Analysis, 183, 2, 215-239, (2007) · Zbl 1105.76013
[11] Holden, H.; Raynaud, X., Global conservative solutions of the Camassa-Holm equation—a Lagrangian point of view, Communications in Partial Differential Equations, 32, 10–12, 1511-1549, (2007) · Zbl 1136.35080
[12] Holden, H.; Raynaud, X., Periodic conservative solutions of the Camassa-Holm equation, Annales de l’Institut Fourier, 58, 3, 945-988, (2008) · Zbl 1158.35079
[13] Holden, H.; Raynaud, X., Global conservative multipeakon solutions of the Camassa-Holm equation, Journal of Hyperbolic Differential Equations, 4, 1, 39-64, (2007) · Zbl 1128.65065
[14] Bressan, A.; Constantin, A., Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5, 1, 1-27, (2007) · Zbl 1139.35378
[15] Tian, L.; Wang, Y.; Zhou, J., Global conservative and dissipative solutions of a coupled Camassa-Holm equations, Journal of Mathematical Physics, 52, (2011) · Zbl 1317.76022
[16] Wang, Y.; Song, Y., On the global existence of dissipative solutions for the modified coupled Camassa-Holm system, Soft Computing, 17, 11, 2007-2019, (2013) · Zbl 06464797
[17] Shen, Z.; Wang, Y.; Karimi, H. R.; Song, Y., On the multipeakon dissipative behavior of the modified coupled Camassa-Holm model for shallow water system, Mathematical Problems in Engineering, 2013, (2013) · Zbl 1296.76019
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