The properties of solutions to the dissipative 2-component Camassa–Holm system

References

• Camassa R, Holm D. An integrable shallow water equation with peaked solitons. Phys. Revi. Lett. 1993;71:1661–1664.
   PubMed Web of Science ®Google Scholar
• Constantin A, Lannes D. The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Rati. Mech. Anal. 2009;192:165–186.
   Web of Science ®Google Scholar
• Constantin A. The trajectories of particles in stokes waves. Inve. Math. 2006;166:523–535.
   Web of Science ®Google Scholar
• Constantin A, Escher J. Analyticity of periodic traveling free surface water waves with vorticity. Anna. Math. 2011;173:559–568.
   Web of Science ®Google Scholar
• Constantin A, Escher J. Particle trajectories in solitary water waves. Bull. Amer. Math. Soci. 2007;44:423–431.
   Web of Science ®Google Scholar
• Constantin A, Strauss W. Stability of peakons. Comm. Pure Appl. Math. 2000;53:603–610.
   Web of Science ®Google Scholar
• Constantin A, Strauss W. Stability of a class of solitary waves in compressible elastic rod. Phys. Lett. A. 2000;270:140–148.
   Web of Science ®Google Scholar
• Yin ZY. On the Cauchy problem for an integrable equation with peakon solutions. Illi. J. Math. 2003;47:649–666.
   Web of Science ®Google Scholar
• Constantin A, McKean HP. A shallow water equation on the circle. Comm. Pure Appl. Math. 1999;52:949–982.
   Web of Science ®Google Scholar
• Constantin A. On the scattering problem for the Camassa-Holm equation. Proc. The Roya. Soci. Lond. (A). 2001;457:953–970.
   Web of Science ®Google Scholar
• Li Y, Olver P. Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Equa. 2000;162:27–63.
   Web of Science ®Google Scholar
• Bressan A, Constantin A. Global conservative solutions of the Camassa-Holm equation. Arch. Rati. Mech. Anal. 2007;183:215–239.
   Web of Science ®Google Scholar
• Bressan A, Constantin A. Global dissipative solutions of the Camassa-Holm equation. Anal. Appl. 2007;5:1–27.
   Web of Science ®Google Scholar
• Constantin A. Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Anna. Inst. Four. (Grenoble). 2000;50:321–362.
   Web of Science ®Google Scholar
• Kong DX. Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves. J. Diff. Equa. 2005;219:421–450.
   Web of Science ®Google Scholar
• Himonas A, Misiolek G, Ponce G, Zhou Y. Persistence properties and unique continuation of solutions of the Camassa-Holm equation. Comm. Math. Phys. 2007;271:511–522.
   Web of Science ®Google Scholar
• Constantin A. Finite propagation speed for the Camassa-Holm equation. J. Math. Phys. 2005;46:023506.
   Web of Science ®Google Scholar
• Henry D. Compactly supported solutions of the Camassa-Holm equation. J. Nonl. Math. Phys. 2005;12:342–347.
   Web of Science ®Google Scholar
• Henry D. Persistence properties for a family of nonlinear partial differential equations. Nonl. Anal. 2009;70:1565–1573.
   Web of Science ®Google Scholar
• Zhou Y, Chen HP. Wave breaking and propagation speed for the Camassa-Holm equation with k=0. Nonl. Anal.: Real Worl. Appl. 2011;12:1875–1882.
   Web of Science ®Google Scholar
• Wu SY, Yin ZY. Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation. J. Diff. Equa. 2009;246:4309–4321.
   Web of Science ®Google Scholar
• Lai SY, Wu YH. Local well-posedness and weak solutions for a weakly dissipative Camassa-Holm equation (in Chinese). Sci. Sin. Math. 2010;40:901–920.
   Google Scholar
• Novruzova E, Hagverdiyev A. On the behavior of the solution of the dissipative Camassa-Holm equation with the arbitrary dispersion coefficient. J. Diff. Equa. 2014;257:4525–4541.
   Web of Science ®Google Scholar
• Chen WX, Tian LX, Deng XY, Zhang JM. Wave breaking for a generalized weakly dissipative 2-component Camassa-Holm system. J. Math. Anal. Appl. 2013;400:406–417.
   Web of Science ®Google Scholar
• Constantin A, Ivanov R. On an integrable 2-component Camassa-Holm shallow water system. Phys. Lett. A. 2008;372:7129–7132.
   Web of Science ®Google Scholar
• Jin YY, Jiang ZH. Wave breaking of an integrable Camassa-Holm system with 2 components. Nonl. Anal. 2014;95:107–116.
   Web of Science ®Google Scholar
• Jin LB, Guo ZG. A note on a modified 2-component Camassa-Holm system. Nonl. Anal.: Real Worl. Appl. 2012;13:887–892.
   Web of Science ®Google Scholar
• Escher J, Lechtenfeld O, Yin ZY. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Disc. Cont. Dyna. Syst. 2007;19:493–513.
   Web of Science ®Google Scholar
• Gui GL, Liu Y. On the global existence and wave-breaking criteria for the 2-component Camassa-Holm system. J. Func. Anal. 2010;258:4251–4278.
   Web of Science ®Google Scholar
• Gui GL, Liu Y. On the Cauchy problem for the 2-component Camassa-Holm system. Math. Zeit. 2011;268:45–46.
   Web of Science ®Google Scholar
• Guo ZG, Ni LD. Persistence properties and unique continuation of solutions to a 2-component Camassa-Holm equation. Math. Phys. Anal. Geom. 2011;14:101–114.
   Web of Science ®Google Scholar
• Shabat A, Alonso L. On the prolongation of a hierarchy of hydrodynamic chains. Nato Scie. Seri. Dord. 2004;132:263–280.
   Google Scholar
• Hu QY. Global existence and blow-up phenomena for a weakly dissipative periodic 2-component Camassa-Holm system. J. Math. Phys. 2011;52:103701.
   Web of Science ®Google Scholar
• Zhu MX, Jiang ZH. Some properties of the solutions to the weakly disspative b-family equation. Nonl. Anal.: Real Worl. Appl. 2012;13:158–167.
   Web of Science ®Google Scholar
• Zong XJ. Properties of the solutions to the 2-component b-family systems. Nonl. Anal. 2012;75:6250–6259.
   Web of Science ®Google Scholar
• Bahouri H, Chemin J, Danchin R. Fourier analysis and nonlinear partial differential equations. Grun. der Math. Wiss: Springer; 2010.
   Google Scholar
• Constantin A, Escher J. Wave breaking for nonlinear nonlocal shallow water equations. Acta. Math. 1998;181:229–243.
   Web of Science ®Google Scholar
• Chen M, Liu SQ, Zhang YJ. A 2-component generalization of the Camassa-Holm equation and its solutions. Lett. Math. Phys. 2006;75:1–15.
   Web of Science ®Google Scholar