Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions

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April 2016, 36(4): 2171-2191. doi: 10.3934/dcds.2016.36.2171

Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions

Yongsheng Mi ^1, , Boling Guo ^2, and Chunlai Mu ^3,

1.	Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
2.	Institute of Applied Physics & Computational Math., Beijing 100088
3.	College of Mathematics and Statistics, Chongqing University, Chongqing 401331

Received February 2015 Revised March 2015 Published September 2015

Abstract
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In this paper, we are concerned with the Cauchy problem of the new integrable four-component system with cubic nonlinearity. We establish the local well-posedness in a range of the Besov spaces. Then the precise blow-up scenario for strong solutions to the system is derived.

Keywords: Besov spaces, local well-posedness., peakon solutions.

Mathematics Subject Classification: 35B30, 35G25, 35A10, 35Q5.

Citation: Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171

References:

[1]	A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl., 5 (2007), 1. doi: 10.1142/S0219530507000857. Google Scholar
[2]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183* (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar*
[3]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[4]	J. Chemin, Localization in Fourier space and Navier-Stokes system. Phase Space Analysis of Partial Differential Equations,, Proceedings, 1 (2004), 53. Google Scholar
[5]	A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166* (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar*
[6]	A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155* (1998), 352. doi: 10.1006/jfan.1997.3231. Google Scholar*
[7]	A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar
[8]	A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7. Google Scholar
[9]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181* (1998), 229. doi: 10.1007/BF02392586. Google Scholar*
[10]	A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa., 26 (1998), 303. Google Scholar
[11]	A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233* (2000), 75. doi: 10.1007/PL00004793. Google Scholar*
[12]	A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197. doi: 10.1088/0266-5611/22/6/017. Google Scholar
[13]	A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group,, Ann. Global Anal. Geom., 31 (2007), 155. doi: 10.1007/s10455-006-9042-8. Google Scholar
[14]	A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Commentarii Mathematici Helvetici, 78 (2003), 787. doi: 10.1007/s00014-003-0785-6. Google Scholar
[15]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192* (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar*
[16]	A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1999), 949. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. Google Scholar
[17]	A. Constantin and W. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar
[18]	R. Danchin, Fourier Analysis Methods for PDEs,, Lecture Notes, (2003). Google Scholar
[19]	R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953. Google Scholar
[20]	A. S. Fokas, The Korteweg-de Vries equation and beyond,, Acta Appl. Math., 39 (1995), 295. doi: 10.1007/BF00994638. Google Scholar
[21]	Y. Fu, G. Gu, Y. Liu and Z. Qu, On the Cauchy problemfor the integrable Camassa- Holm type equation with cubic nonlinearity,, , (): 1. Google Scholar
[22]	B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation,, Physica D, 95 (1996), 229. doi: 10.1016/0167-2789(96)00048-6. Google Scholar
[23]	G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,, J. Funct. Anal., 258 (2010), 4251. doi: 10.1016/j.jfa.2010.02.008. Google Scholar
[24]	H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 24 (2009), 1047. doi: 10.3934/dcds.2009.24.1047. Google Scholar
[25]	H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equations-a Lagrangianpoiny of view,, Comm. Partial Differential Equations, 32 (2007), 1511. doi: 10.1080/03605300601088674. Google Scholar
[26]	S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690. Google Scholar
[27]	J. Lenells, A variational approach to the stability of periodic peakons,, J. Nonlinear Math. Phys., 11 (2004), 151. doi: 10.2991/jnmp.2004.11.2.2. Google Scholar
[28]	Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Differential Equations, 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar
[29]	G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar
[30]	P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900. Google Scholar
[31]	Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons,, J. Math. Phys., 47 (2006). doi: 10.1063/1.2365758. Google Scholar
[32]	Z. Qiao and X. Li, An integrable equation with nonsmooth solitons,, Theor. Math. Phys., 167* (2011), 584. doi: 10.1007/s11232-011-0044-8. Google Scholar*
[33]	J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1. Google Scholar
[34]	B. Xia and Z. Qiao, Integrable multi-component Camassa-Holm system,, , (2013). Google Scholar

show all references

References:

[1]	A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl., 5 (2007), 1. doi: 10.1142/S0219530507000857. Google Scholar
[2]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183* (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar*
[3]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[4]	J. Chemin, Localization in Fourier space and Navier-Stokes system. Phase Space Analysis of Partial Differential Equations,, Proceedings, 1 (2004), 53. Google Scholar
[5]	A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166* (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar*
[6]	A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155* (1998), 352. doi: 10.1006/jfan.1997.3231. Google Scholar*
[7]	A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar
[8]	A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7. Google Scholar
[9]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181* (1998), 229. doi: 10.1007/BF02392586. Google Scholar*
[10]	A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa., 26 (1998), 303. Google Scholar
[11]	A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233* (2000), 75. doi: 10.1007/PL00004793. Google Scholar*
[12]	A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197. doi: 10.1088/0266-5611/22/6/017. Google Scholar
[13]	A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group,, Ann. Global Anal. Geom., 31 (2007), 155. doi: 10.1007/s10455-006-9042-8. Google Scholar
[14]	A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Commentarii Mathematici Helvetici, 78 (2003), 787. doi: 10.1007/s00014-003-0785-6. Google Scholar
[15]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192* (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar*
[16]	A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1999), 949. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. Google Scholar
[17]	A. Constantin and W. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar
[18]	R. Danchin, Fourier Analysis Methods for PDEs,, Lecture Notes, (2003). Google Scholar
[19]	R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953. Google Scholar
[20]	A. S. Fokas, The Korteweg-de Vries equation and beyond,, Acta Appl. Math., 39 (1995), 295. doi: 10.1007/BF00994638. Google Scholar
[21]	Y. Fu, G. Gu, Y. Liu and Z. Qu, On the Cauchy problemfor the integrable Camassa- Holm type equation with cubic nonlinearity,, , (): 1. Google Scholar
[22]	B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation,, Physica D, 95 (1996), 229. doi: 10.1016/0167-2789(96)00048-6. Google Scholar
[23]	G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,, J. Funct. Anal., 258 (2010), 4251. doi: 10.1016/j.jfa.2010.02.008. Google Scholar
[24]	H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 24 (2009), 1047. doi: 10.3934/dcds.2009.24.1047. Google Scholar
[25]	H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equations-a Lagrangianpoiny of view,, Comm. Partial Differential Equations, 32 (2007), 1511. doi: 10.1080/03605300601088674. Google Scholar
[26]	S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690. Google Scholar
[27]	J. Lenells, A variational approach to the stability of periodic peakons,, J. Nonlinear Math. Phys., 11 (2004), 151. doi: 10.2991/jnmp.2004.11.2.2. Google Scholar
[28]	Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Differential Equations, 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar
[29]	G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar
[30]	P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900. Google Scholar
[31]	Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons,, J. Math. Phys., 47 (2006). doi: 10.1063/1.2365758. Google Scholar
[32]	Z. Qiao and X. Li, An integrable equation with nonsmooth solitons,, Theor. Math. Phys., 167* (2011), 584. doi: 10.1007/s11232-011-0044-8. Google Scholar*
[33]	J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1. Google Scholar
[34]	B. Xia and Z. Qiao, Integrable multi-component Camassa-Holm system,, , (2013). Google Scholar

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