Nonuniform dependence and well-posedness for the generalized Camassa–Holm equation

References

• Novikov V . Generalization of the Camassa–Holm equation. J Phys A. 2009;42: 342002:1–14.
   Google Scholar
• Tu X , Yin Z . Local well-posedness and blow-up phenomena for a generalized Camassa–Holm equation with peakon solutions. Discrete Contin Dyn Syst A. 2016;128:1–19.
   Google Scholar
• Tu X , Yin Z . Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation in the critical Besov space. Nonlinear Anal TMA. 2015;128:1–19.
   Web of Science ®Google Scholar
• Tu X , Yin Z [cited 2016 Jan 5]. Global weak solutions for a generalized Camassa–Holm equation. 2016. arXiv:1511.02848v2 [math.AP].
   Google Scholar
• Tu X , Yin Z [cited 2015 Nov 7]. Analyticity of the Cauchy problem and persistence properties for a generalized Camassa–Holm equation. 2016. arXiv:1511.02316v1 [math.AP].
   Google Scholar
• Geng X , Xue B . An extension of integrable peakon equations with cubic nonlinearity. Nonlinearity. 2009;22:1847–56.
   Web of Science ®Google Scholar
• Fokas A , Fuchssteiner B . Symplectic structures, their Backlund transformation and hereditary symmetries. Physica D. 1981;4:47–66.
   Web of Science ®Google Scholar
• Camassa R , Holm D . An integrable shallow water equation with peaked solitons. Phys Rev Lett. 1993;71:1661–1664.
   PubMed Web of Science ®Google Scholar
• Constantin A . On the scattering problem for the Camassa–Holm equation. Proc R Soc Lond. A. 2001;457:953–970.
   Web of Science ®Google Scholar
• Constantin A . The trajectories of particles in Stokes waves. Invent Math. 2006;166:523–535.
   Web of Science ®Google Scholar
• Constantin A , Escher J . Particle trajectores in solitary water waves. Bull Amer Math Soc. 2007;44:423–431.
   Web of Science ®Google Scholar
• Constantin A , Strauss W . Stability of peakons. Commun Pure Appl Math. 2000;53:603–610.
   Web of Science ®Google Scholar
• Constantin A , Escher J . Analyticity of periodic traveling free surface water waves with vorticity. Ann Math. 2011;173:559–568.
   Web of Science ®Google Scholar
• Amick CJ , Fraenkel LE , Toland JF . On the Stokes conjecture for the wave of extreme form. Acta Math. 1982;148:193–214.
   Web of Science ®Google Scholar
• Constantin A , McKean HP . A shallow water equation on the circle. Commun Pure Appl Math. 1999;52:949–982.
   Web of Science ®Google Scholar
• Constantin A . On the inverse spectral problem for the Camassa–Holm equation. J Funct Anal. 1998;155:352–363.
   Web of Science ®Google Scholar
• Constantin A , Gerdjikov V , Ivanov RI . Inverse scattering transform for the Camassa–Holm equation. Inverse Probl. 2006;22:2197–2207.
   Web of Science ®Google Scholar
• Constantin A , Kappeler T , Kolev B , et al . On Geodesic exponential maps of the Virasoro group. Ann Global Anal Geom. 2007;31:155–180.
   Web of Science ®Google Scholar
• Misiolek GA . Shallow water equation as a geodesic flow on the Bott–Virasoro group. J Geom Phys. 1998;24:203–208.
   Web of Science ®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
• Bressan A , Constantin A . Global conservative solutions of the Camassa–Holm equation. Arch Ration Mech Anal. 2007;183:215–239.
   Web of Science ®Google Scholar
• Holden H , Raynaud X . Global conservative solutions of the Camassa–Holm equations-a Lagrangianpoiny of view. Commun Partial Differ Equ. 2007;32:1511–1549.
   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
• Holden H , Raynaud X . Dissipative solutions for the Camassa–Holm equation. Discrete Contin Dyn Syst. 2009;24:1047–1112.
   Web of Science ®Google Scholar
• Lakshmanan M . Integrable nonlinear wave equations and possible connections to tsunami dynamics. In: Lakshmanan M , editor. Tsunami and nonlinear waves. Berlin: Springer; 2007. p. 31–49.
   Google Scholar
• Segur H . Integrable models of waves in shallow water. In: Segur H , editor. Probability, geometry and integrable systems. Vol. 55, MSRI publications. Cambridge: Cambridge University Press; 2008. p. 345–371.
   Google Scholar
• Segur H . Waves in shallow water with emphasis on the tsunami of 2004. In: Segur H , editor. Tsunami and nonlinear waves. Berlin: Springer; 2007. p. 3–29.
   Google Scholar
• Constantin A , Johnson RS . Propagation of very long water waves with vorticity, over variable depth, with applications to tsunamis. Fluid Dynam Res. 2008;40:175–211.
   Web of Science ®Google Scholar
• Himonas A , Mantzavinos D . The Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation. Nonlinear Appl. 2014;95:499–529.
   Web of Science ®Google Scholar
• Himonas A , Mantzavinos D . The Cauchy problem for a 4-parameter family of equations with peakon traveling waves. Nonlinear Anal. 2016;133:161–199.
   Web of Science ®Google Scholar
• Degasperis A , Procesi M . Asymptotic integrability. Symmetry Perturbation Theory. 1999;1(1):23–37.
   Google Scholar
• Degasperis A , Holm DD , Hone ANW . A new integrable equation with peakon solutions. Theor Math Phys. 2002;133:1461–72.
   Web of Science ®Google Scholar
• Constantin A , Ivanov R , Lenells J . Inverse scattering transform for the Degasperis–Procesi equation. Nonlinearity. 2010;23:2559–2575.
   Web of Science ®Google Scholar
• Degasperis A , Procesi M . Asymptotic integrability. In: Degasperis A , Gaeta G , editors. Symmetry and perturbation theory. Singapore: World Scientific; 1999. p. 23–37.
   Google Scholar
• Toland JF . Stokes waves. Topol Methods Nonlinear Anal. 1996;7:1–48.
   Google Scholar
• Holm DD , Staley MF . Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J Appl Dyn Syst. 2003;2:323–380.
   Web of Science ®Google Scholar
• Lundmark H , Szmigielski J . Multi-peakon solutions of the Degasperis–Procesi equation. Inverse Probl. 2005;21:1553–1570.
   Web of Science ®Google Scholar
• Matsuno Y . Multisoliton solutions of the Degasperis–Procesi equation and their peakon limit. Inverse Probl. 2003;19:1241–1245.
   Web of Science ®Google Scholar
• Lenells J . Traveling wave solutions of the Degasperis–Procesi equation. J Math Anal Appl. 2005;306:72–82.
   Web of Science ®Google Scholar
• Vakhnenko VO , Parkes EJ . Periodic and solitary-wave solutions of the Degasperis–Procesi equation. Chaos Solitons Fractals. 2004;20:1059–1073.
   Web of Science ®Google Scholar
• Home ANW , Wang JP . Integrable peakon equations with cubic nonlinearity. J Phys A. 2008;41: 372002:1–11.
   Google Scholar
• Wu X , Yin Z . Well-posedness and global existence for the Novikov equation. Ann Sc Norm Super Pisa Cl Sci. 2012;11:707–727.
   Google Scholar
• Ni L , Zhou Y . Well-posedness and persistence properties for the Novikov equation. J Differ Equ. 2011;250:3002–3021.
   Web of Science ®Google Scholar
• Wu X , Yin Z . A note on the Cauchy problem of the Novikov equation. Appl Anal. 2013;92:1116–1137.
   Web of Science ®Google Scholar
• Himonas A , Holliman C . The Cauchy problem for the Novikov equation. Nonlinearity. 2012;25:449–479.
   Web of Science ®Google Scholar
• Jiang Z , Ni L . Blow-up phenomenon for the integrable Novikov equation. J Math Anal Appl. 2012;385:551–558.
   Web of Science ®Google Scholar
• Yan W , Li Y , Zhang Y . Global existence and blow-up phenomena for the weakly dissipative Novikov equation. Nonlinear Anal. 2012;75:2464–2473.
   Web of Science ®Google Scholar
• Hone W , Lundmark H , Szmigielski J . Explicit multipeakon solutions of Novikov cubically nonlinear integrable Camassa-Holm type equation. Dyn Partial Differ Equ. 2009;6:253–289.
   Web of Science ®Google Scholar
• Himonas A , Misiołek G . High-frequency smooth solutions and well-posedness of the Camassa–Holm equation. Int Math Res Not. 2005;51:3135–3151.
   Google Scholar
• Himonas A , Kenig C . Non-uniform dependence on initial data for the CH equation on the line. Differ Integral Equ. 2009;22:201–224.
   Web of Science ®Google Scholar
• Himonas A , Kenig C , Misiołek G . Non-uniform dependence for the periodic CH equation. Commun Partial Differ Equ. 2010;35:1145–1162.
   Web of Science ®Google Scholar
• Christov O , Hakkaev S . On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis–Procesi equation. J Math Anal Appl. 2009;360:47–56.
   Web of Science ®Google Scholar
• Himonas A , Holliman C . On well-posedness of the Degasperis–Procesi equation. Discrete Contin Dyn Syst. 2011;31:469–488.
   Web of Science ®Google Scholar
• Holliman C . Non-uniform dependence and well-posedness for the periodic Hunter–Saxton equation. Differ Int Equ. 2010;23:1150–1194.
   Google Scholar
• Himonas A , Misiołek G . Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics. Commun Math Phys. 2009;296:285–301.
   Web of Science ®Google Scholar
• Taylor M . Pseudodifferential operators and nonlinear PDE. Boston: Birkhäuser; 1991.
   Google Scholar
• Taylor M . Partial differential equations, III: nonlinear equations. New york (NY): Springer; 1996.
   Google Scholar
• Dieudonné J . Foundations of modern analysis. New York (NY): Academic Press; 1969.
   Google Scholar