References
- Xia B, Qiao Z, Zhou R. A synthetical integrable two-component model with peakon solutions. arXiv:1301.3216v1 [nlin.SI], 2013 Jan 15.
- Camassa , R. and Holm , D. 1993 . An integrable shallow water equation with peaked solitons . Physical Review Letters , 71 : 1661 – 1664 .
- Constantin , A. and Lannes , D. 2009 . The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations . Archive for Rational Mechanics and Analysis , 192 : 165 – 186 .
- Constantin , A. 2006 . The trajectories of particles in Stokes waves . Inventiones Mathematicae , 166 : 523 – 535 .
- Toland , J.F. 1996 . Stokes waves . Topological Methods in Nonlinear Analysis , 7 : 1 – 48 .
- Constantin , A. and Escher , J. 2007 . Particle trajectories in solitary water waves . Bulletin of the American Mathematical Society , 44 : 423 – 431 .
- Constantin , A. and Strauss , W. 2000 . Stability of peakons . Communications on Pure and Applied Mathematics , 53 : 603 – 610 .
- Lenells , J. 2004 . A variational approach to the stability of periodic peakons . Journal of Nonlinear Mathematical Physics , 11 : 151 – 163 .
- Constantin , A. and McKean , H.P. 1999 . A shallow water equation on the circle . Communications on Pure and Applied Mathematics , 52 : 949 – 982 .
- Constantin , A. , Gerdjikov , V. and Ivanov , R. 2006 . Inverse scattering transform for the Camassa–Holm equation . Inverse Problems , 22 : 2197 – 2207 .
- Constantin , A. 1998 . On the inverse spectral problem for the Camassa–Holm equation . Journal of Functional Analysis , 155 : 352 – 363 .
- Constantin , A. , Kappeler , T. , Kolev , B. and Topalov , P. 2007 . On geodesic exponential maps of the Virasoro group . Annals of Global Analysis and Geometry , 31 : 155 – 180 .
- Constantin , A. and Kolev , B. 2003 . Geodesic flow on the diffeomorphism group of the circle . Commentarii Mathematici Helvetici , 78 : 787 – 804 .
- Kouranbaeva , S. 1999 . The Camassa–Holm equation as a geodesic flow on the diffeomorphism group . Journal of Mathematical Physics , 40 : 857 – 868 .
- Misiolek , G. 1998 . A shallow water equation as a geodesic flow on the Bott–Virasoro group . Journal of Geometry and Physics , 24 : 203 – 208 .
- Constantin , A. 2000 . Existence of permanent and breaking waves for a shallow water equation: a geometric approach . Annales de l’Institut Fourier (Grenoble) , 50 : 321 – 362 .
- Constantin , A. and Escher , J. 1998 . Wave breaking for nonlinear nonlocal shallow water equations . Acta Mathematica , 181 : 229 – 243 .
- Constantin , A. and Escher , J. 2000 . On the blow-up rate and the blow-up set of breaking waves for a shallow water equation . Mathematische Zeitschrift , 233 : 75 – 91 .
- Constantin , A. and Escher , J. 1998 . Global existence and blow-up for a shallow water equation . Annali della Scuola Normale Superiore di Pisa , 26 : 303 – 328 .
- Li , Y. and Olver , P. 2000 . Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation . Journal of Differential Equations , 162 : 27 – 63 .
- Bressan , A. and Constantin , A. 2007 . Global conservative solutions of the Camassa–Holm equation . Archive for Rational Mechanics and Analysis , 183 : 215 – 239 .
- Holden , H. and Raynaud , X. 2007 . Global conservative solutions of the Camassa–Holm equations – a Lagrangian point of view . Communications in Partial Differential Equations , 32 : 1511 – 1549 .
- Bressan , A. and Constantin , A. 2007 . Global dissipative solutions of the Camassa–Holm equation . Applied Analysis , 5 : 1 – 27 .
- Holden , H. and Raynaud , X. 2009 . Dissipative solutions for the Camassa–Holm equation . Discrete and Continuous Dynamical Systems , 24 : 1047 – 1112 .
- Novikov V. Generalizations of the Camassa–Holm equation. Journal of Physics A. 2009;42:342002 1–15.
- Hone , A. , Lundmark , H. and Szmigielski , J. 2009 . Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa–Holm type equation . Dynamics of Partial Differential Equations , 6 : 253 – 289 .
- Hone A, Wang J. Integrable peakon equations with cubic nonlinearity. Journal of Physics A. 2008;41:372002 1–19.
- Tiglay , F. 2005 . The periodic Cauchy problem of the modified Hunter–Saxton equation . Journal of Evolution Equations , 5 : 509 – 527 .
- Tiglay , F. 2011 . The periodic Cauchy problem for Novikov’s equation . International Mathematics Research Notices , 2011 : 4633 – 4648 .
- Gorka , P. and Reyes , E. 2011 . The modified Camassa–Holm equation . International Mathematics Research Notices , 2011 : 2617 – 2649 .
- Fu , Y. , Liu , Y. and Qu , C. 2010 . Well-posedness and blow-up solution for a modified twocomponent periodic Camassa–Holm system with peakons . Mathematische Annalen , 348 : 415 – 448 .
- Gui , G. and Liu , Y. 2010 . On the global existence and wave-breaking criteria for the two-component Camassa–Holm system . Journal of Functional Analysis , 258 : 4251 – 4278 .
- Gui , G. and Liu , Y. 2011 . On the Cauchy problem for the two-component Camassa–Holm system . Mathematische Zeitschrift , 268 : 45 – 66 .
- Holm D, Ivanov R. Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples. Journal of Physics A. 2010;43:492001 1–21.
- Holm D, Ivanov R. Two-component CH system: inverse scattering, peakons and geometry. Inverse Problems. 2011;27:045013 1–21.
- Holm D, Onaraigh L, Tronci C. Singular solutions of a modified two-component Camassa–Holm equation. Physical Review E. 2009;79:016601 1–9.
- Constantin , A. and Ivanov , R. 2008 . On an integrable two-component Camassa–Holm shallow water system . Physics Letters A , 372 : 7129 – 7132 .
- Danchin , R. 2001 . A few remarks on the Camassa–Holm equation . Differential Integral Equations , 14 : 953 – 988 .
- Constantin , A. and Escher , J. 2011 . Analyticity of periodic traveling free surface water waves with vorticity . Annals of Mathematics , 178 : 559 – 568 .
- Chemin J. Localization in Fourier space and Navier-Stokes system. Phase Space Analysis of Partial Differential Equations. Proceedings, CRM series, Pisa; 2004. p. 53–136.
- Danchin R. Fourier analysis methods for PDEs, Lecture Notes, 2003 Nov 14.
- Himonas , A. and Misiolek , G. 2003 . Analyticity of the Cauchy problem for an integrable evolution equation . Mathematische Annalen , 327 : 575 – 584 .
- Baouendi , M. and Goulaouic , C. 1983 . Sharp estimates for analytic pseudodifferential operators and application to the Cauchy problems . Journal of Differential Equations , 48 : 241 – 268 .