October  2012, 32(10): 3459-3484. doi: 10.3934/dcds.2012.32.3459

Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system

1. 

Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408, United States

2. 

Department of Mathematics and Physics, Huaiyin Institute of Technology, Huaiyin, Jiangsu 223003, China

Received  April 2011 Revised  January 2012 Published  May 2012

Considered herein is the generalized two-component periodic Camassa-Holm system. The precise blow-up scenarios of strong solutions and several results of blow-up solutions with certain initial profiles are described in detail. The exact blow-up rates are also determined. Finally, a sufficient condition for global solutions is established.
Citation: Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459
References:
[1]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Archive for Rational Mechanics and Analysis, 183 (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar

[2]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Analysis and its Applications (Singap.), 5 (2007), 1. Google Scholar

[3]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation,, Journal of Differential Equations, 141 (1997), 218. doi: 10.1006/jdeq.1997.3333. Google Scholar

[4]

A. Constantin, On the blow-up of solutions of a periodic shallow water equation,, Journal of Nonlinear Science, 10 (2000), 391. doi: 10.1007/s003329910017. Google Scholar

[5]

A. Constantin, The trajectories of particles in Stokes waves,, Inventiones Mathematicae, 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar

[6]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis,, Fluid Dynamics Research, 40 (2008), 175. doi: 10.1016/j.fluiddyn.2007.06.004. Google Scholar

[7]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Mathematica, 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar

[8]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomenon for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar

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A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar

[10]

A. Constantin and H. P. Mckean, A shallow water equation on the circle,, Communications on Pure and Applied Mathematics, 52 (1999), 949. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. Google Scholar

[11]

A. Constantin and W. A. Strauss, Stability of peakons,, Communications on Pure and Applied Mathematics, 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C. Google Scholar

[12]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, Journal of Nonlinear Science, 12 (2002), 415. doi: 10.1007/s00332-002-0517-x. Google Scholar

[13]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries,, Phys. D., 4 (): 47. Google Scholar

[14]

A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow water equation,, Differential Integral Equations, 14 (2001), 821. Google Scholar

[15]

A. Shabat and L. Martínez Alonso, On the prolongation of a hierarchy of hydrodynamic chains,, in, 132 (2004), 263. Google Scholar

[16]

C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system,, Journal of Differential Equations, 248 (2010), 2003. doi: 10.1016/j.jde.2009.08.002. Google Scholar

[17]

C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,, Communications on Pure and Applied Mathematics, 46 (1993), 527. doi: 10.1002/cpa.3160460405. Google Scholar

[18]

D. Henry, Infinite propagation speed for a two component Camassa-Holm equation,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597. doi: 10.3934/dcdsb.2009.12.597. Google Scholar

[19]

G. B. Whitham, "Linear and Nonlinear Waves,", Reprint of the 1974 original, (1974). Google Scholar

[20]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,, Journal of Functional Analysis, 258 (2010), 4251. doi: 10.1016/j.jfa.2010.02.008. Google Scholar

[21]

G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system,, Mathematische Zeitschrift, 268 (2011), 45. doi: 10.1007/s00209-009-0660-2. Google Scholar

[22]

G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080. doi: 10.1007/PL00012648. Google Scholar

[23]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar

[24]

H. Aratyn, J. F. Gomes and A. H. Zimerman, On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa-Holm equation,, SIGMA Symmetry, 2 (2006). Google Scholar

[25]

H. P. Mckean, Breakdown of a shallow water equation,, Asian Journal of Mathematics, 2 (1998), 867. Google Scholar

[26]

J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 19 (2007), 493. Google Scholar

[27]

J. F. Toland, Stokes waves,, Topological Methods in Nonlinear Analysis, 7 (1996), 1. Google Scholar

[28]

J. Lenells, Stability of periodic peakons,, International Mathematics Research Notices, 2004 (): 485. Google Scholar

[29]

M. Chen, S.-Q. Liu and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions,, Letters in Mathematical Physics, 75 (2006), 1. doi: 10.1007/s11005-005-0041-7. Google Scholar

[30]

O. Mustafa, On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system,, Wave Motion, 46 (2009), 397. doi: 10.1016/j.wavemoti.2009.06.011. Google Scholar

[31]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E (3), 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900. Google Scholar

[32]

Q. Hu and Z. Yin, Blowup phenomena for a new periodic nonlinearly dispersive wave equation,, Math. Nachr., 283 (2010), 1613. doi: 10.1002/mana.200810075. Google Scholar

[33]

R. Beals, D. Sattinger and J. Szmigielski, Multi-peakons and a theorem of Stieltjes,, Inverse Problems, 15 (1999). doi: 10.1088/0266-5611/15/1/001. Google Scholar

[34]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Physical Review Letters, 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar

[35]

R. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case,, Wave Motion, 46 (2009), 389. doi: 10.1016/j.wavemoti.2009.06.012. Google Scholar

[36]

R. M. Chen and Y. Liu, Wave-breaking and global existence for a generalized two-component Camassa-Holm system,, Int. Math. Res. Not., 2011 (): 1381. Google Scholar

[37]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in, 448 (1975), 25. Google Scholar

[38]

T. Tao, Low-regularity global solutions to nonlinear dispersive equations,, in, 40 (2002), 19. Google Scholar

[39]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation,, J. Differential Equations, 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar

[40]

Y. Liu and P. Zhang, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,, International Mathematics Research Notices, 2010 (): 1981. Google Scholar

[41]

Z. Popowicz, A 2-component or N=2 supersymmetric Camassa-Holm equation,, Physics Letters A, 354 (2006), 110. doi: 10.1016/j.physleta.2006.01.027. Google Scholar

[42]

Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation,, Dyn. Cont. Discrete Impuls. Syst. Ser. A Math. Anal., 12 (2005), 375. Google Scholar

show all references

References:
[1]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Archive for Rational Mechanics and Analysis, 183 (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar

[2]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Analysis and its Applications (Singap.), 5 (2007), 1. Google Scholar

[3]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation,, Journal of Differential Equations, 141 (1997), 218. doi: 10.1006/jdeq.1997.3333. Google Scholar

[4]

A. Constantin, On the blow-up of solutions of a periodic shallow water equation,, Journal of Nonlinear Science, 10 (2000), 391. doi: 10.1007/s003329910017. Google Scholar

[5]

A. Constantin, The trajectories of particles in Stokes waves,, Inventiones Mathematicae, 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar

[6]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis,, Fluid Dynamics Research, 40 (2008), 175. doi: 10.1016/j.fluiddyn.2007.06.004. Google Scholar

[7]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Mathematica, 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar

[8]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomenon for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar

[9]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar

[10]

A. Constantin and H. P. Mckean, A shallow water equation on the circle,, Communications on Pure and Applied Mathematics, 52 (1999), 949. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. Google Scholar

[11]

A. Constantin and W. A. Strauss, Stability of peakons,, Communications on Pure and Applied Mathematics, 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C. Google Scholar

[12]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, Journal of Nonlinear Science, 12 (2002), 415. doi: 10.1007/s00332-002-0517-x. Google Scholar

[13]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries,, Phys. D., 4 (): 47. Google Scholar

[14]

A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow water equation,, Differential Integral Equations, 14 (2001), 821. Google Scholar

[15]

A. Shabat and L. Martínez Alonso, On the prolongation of a hierarchy of hydrodynamic chains,, in, 132 (2004), 263. Google Scholar

[16]

C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system,, Journal of Differential Equations, 248 (2010), 2003. doi: 10.1016/j.jde.2009.08.002. Google Scholar

[17]

C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,, Communications on Pure and Applied Mathematics, 46 (1993), 527. doi: 10.1002/cpa.3160460405. Google Scholar

[18]

D. Henry, Infinite propagation speed for a two component Camassa-Holm equation,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597. doi: 10.3934/dcdsb.2009.12.597. Google Scholar

[19]

G. B. Whitham, "Linear and Nonlinear Waves,", Reprint of the 1974 original, (1974). Google Scholar

[20]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,, Journal of Functional Analysis, 258 (2010), 4251. doi: 10.1016/j.jfa.2010.02.008. Google Scholar

[21]

G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system,, Mathematische Zeitschrift, 268 (2011), 45. doi: 10.1007/s00209-009-0660-2. Google Scholar

[22]

G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080. doi: 10.1007/PL00012648. Google Scholar

[23]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar

[24]

H. Aratyn, J. F. Gomes and A. H. Zimerman, On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa-Holm equation,, SIGMA Symmetry, 2 (2006). Google Scholar

[25]

H. P. Mckean, Breakdown of a shallow water equation,, Asian Journal of Mathematics, 2 (1998), 867. Google Scholar

[26]

J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 19 (2007), 493. Google Scholar

[27]

J. F. Toland, Stokes waves,, Topological Methods in Nonlinear Analysis, 7 (1996), 1. Google Scholar

[28]

J. Lenells, Stability of periodic peakons,, International Mathematics Research Notices, 2004 (): 485. Google Scholar

[29]

M. Chen, S.-Q. Liu and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions,, Letters in Mathematical Physics, 75 (2006), 1. doi: 10.1007/s11005-005-0041-7. Google Scholar

[30]

O. Mustafa, On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system,, Wave Motion, 46 (2009), 397. doi: 10.1016/j.wavemoti.2009.06.011. Google Scholar

[31]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E (3), 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900. Google Scholar

[32]

Q. Hu and Z. Yin, Blowup phenomena for a new periodic nonlinearly dispersive wave equation,, Math. Nachr., 283 (2010), 1613. doi: 10.1002/mana.200810075. Google Scholar

[33]

R. Beals, D. Sattinger and J. Szmigielski, Multi-peakons and a theorem of Stieltjes,, Inverse Problems, 15 (1999). doi: 10.1088/0266-5611/15/1/001. Google Scholar

[34]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Physical Review Letters, 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar

[35]

R. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case,, Wave Motion, 46 (2009), 389. doi: 10.1016/j.wavemoti.2009.06.012. Google Scholar

[36]

R. M. Chen and Y. Liu, Wave-breaking and global existence for a generalized two-component Camassa-Holm system,, Int. Math. Res. Not., 2011 (): 1381. Google Scholar

[37]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in, 448 (1975), 25. Google Scholar

[38]

T. Tao, Low-regularity global solutions to nonlinear dispersive equations,, in, 40 (2002), 19. Google Scholar

[39]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation,, J. Differential Equations, 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar

[40]

Y. Liu and P. Zhang, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,, International Mathematics Research Notices, 2010 (): 1981. Google Scholar

[41]

Z. Popowicz, A 2-component or N=2 supersymmetric Camassa-Holm equation,, Physics Letters A, 354 (2006), 110. doi: 10.1016/j.physleta.2006.01.027. Google Scholar

[42]

Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation,, Dyn. Cont. Discrete Impuls. Syst. Ser. A Math. Anal., 12 (2005), 375. Google Scholar

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