Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: 35B30, 35G25, 35A10, 35Q53.


    \begin{equation} \\ \end{equation}
  • [1]

    A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.doi: 10.1142/S0219530507000857.


    A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.doi: 10.1007/s00205-006-0010-z.


    R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.doi: 10.1103/PhysRevLett.71.1661.


    J. Chemin, Localization in Fourier space and Navier-Stokes system. Phase Space Analysis of Partial Differential Equations, Proceedings, CRM series, Pisa, 1 (2004), 53-135.


    A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.doi: 10.1007/s00222-006-0002-5.


    A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.doi: 10.1006/jfan.1997.3231.


    A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.doi: 10.5802/aif.1757.


    A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.doi: 10.1090/S0273-0979-07-01159-7.


    A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.doi: 10.1007/BF02392586.


    A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa., 26 (1998), 303-328.


    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), 75-91.doi: 10.1007/PL00004793.


    A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.doi: 10.1088/0266-5611/22/6/017.


    A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180.doi: 10.1007/s10455-006-9042-8.


    A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Commentarii Mathematici Helvetici, 78 (2003), 787-804.doi: 10.1007/s00014-003-0785-6.


    A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.doi: 10.1007/s00205-008-0128-2.


    A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.


    A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.


    R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14 November, 2003.


    R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.


    A. S. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305.doi: 10.1007/BF00994638.


    Y. Fu, G. Gu, Y. Liu and Z. Qu, On the Cauchy problemfor the integrable Camassa- Holm type equation with cubic nonlinearity, arXiv:1108.5368v2, 1-27.


    B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 229-243.doi: 10.1016/0167-2789(96)00048-6.


    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), 4251-4278.doi: 10.1016/j.jfa.2010.02.008.


    H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112.doi: 10.3934/dcds.2009.24.1047.


    H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equations-a Lagrangianpoiny of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549.doi: 10.1080/03605300601088674.


    S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868.doi: 10.1063/1.532690.


    J. Lenells, A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163.doi: 10.2991/jnmp.2004.11.2.2.


    Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.doi: 10.1006/jdeq.1999.3683.


    G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.doi: 10.1016/S0393-0440(97)00010-7.


    P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.doi: 10.1103/PhysRevE.53.1900.


    Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9pp.doi: 10.1063/1.2365758.


    Z. Qiao and X. Li, An integrable equation with nonsmooth solitons, Theor. Math. Phys., 167 (2011), 584-589.doi: 10.1007/s11232-011-0044-8.


    J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.


    B. Xia and Z. Qiao, Integrable multi-component Camassa-Holm system, arXiv:1310.0268v1 [nlin.SI] 1 Oct 2013.

  • 加载中

Article Metrics

HTML views() PDF downloads(153) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint