On a generalized Camassa–Holm equation with the flow generated by velocity and its gradient

References

• Novikov V. Generalization of the Camassa--Holm equation. J. Phys. A. 2009;42:342002 14 pp.
   Google Scholar
• Camassa R, Holm DD. An integrable shallow water equation with peaked solitons. Phys. Rev. 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. Ration. Mech. Anal. 2009;192:165–186.
   Web of Science ®Google Scholar
• Ionescu-Kruse D. Variational derivation of the Camassa--Holm shallow water equation. J. Nonlinear Math. Phys. 2007;14:303–312.
   Web of Science ®Google Scholar
• Constantin A. On the scattering problem for the Camassa--Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 2001;457:953–970.
   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. The Hamiltonian structure of the Camassa--Holm equation. Exposition Math. 1997;15:53–85.
   Google Scholar
• Fokas A, Fuchssteiner B. Symplectic structures, their B\"{a}cklund transformation and hereditary symmetries. Phys. D. 1981/82;4:47–66.
   Web of Science ®Google Scholar
• Constantin A, Strauss WA. Stability of peakons. Commun. Pure Appl. Math. 2000;53:603–610.
   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. Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math (2). 2011;173:559–568.
   Web of Science ®Google Scholar
• Toland JF. Stokes waves. Topological Methods Nonlinear Anal. 1996;7:1–48.
   Google Scholar
• Constantin A, Escher J. Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci (4). 1998;26:303–328.
   Google Scholar
• Constantin A, Escher J. Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 1998;51:475–504.
   Web of Science ®Google Scholar
• Danchin R. A few remarks on the Camassa--Holm equation. Differ. Integral Equ. 2001;14:953–988.
   Google Scholar
• Rodriguez-Blanco G. On the Cauchy problem for the Camassa--Holm equation. Nonlinear Anal. Ser. A: Theory Methods. 2001;46:309–327.
   Web of Science ®Google Scholar
• Constantin A. Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble). 2000;50:321–362.
   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
• Constantin A, Molinet L. Global weak solutions for a shallow water equation. Commun. Math. Phys. 2000;211:45–61.
   Web of Science ®Google Scholar
• Xin Z, Zhang P. On the weak solutions to a shallow water equation. Commun. Pure Appl. Math. 2000;53:1411–1433.
   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
• Bressan A, Constantin A. Global dissipative solutions of the Camassa--Holm equation. Anal. Appl. (Singapore). 2007;5:1–27.
   Web of Science ®Google Scholar
• Degasperis A, Procesi M. Asymptotic integrability. Symmetry and perturbation theory (Rome, 1998). River Edge (NJ): World Scientific Publishing, 1998, pp. 23–37.
   Google Scholar
• Degasperis A, Holm DD, Hone ANW. A new integrable equation with peakon solutions. Theoret. Math. Phys. 2002;133:1463–1474.
   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
• Dullin HR, Gottwald GA, Holm DD. On asymptotically equivalent shallow water wave equations. Phys. D. 2004;190:1–14.
   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
• Gui G, Liu Y. On the Cauchy problem for the Degasperis--Procesi equation. Quart. Appl. Math. 2011;69:445–464.
   Web of Science ®Google Scholar
• Himonas AA, Holliman C. The Cauchy problem for the Novikov equation. Nonlinearity. 2012;25:449–479.
   Web of Science ®Google Scholar
• Yin Z. Global existence for a new periodic integrable equation. J. Math. Anal. Appl. 2003;283:129–139.
   Web of Science ®Google Scholar
• Liu Y, Yin Z. Global existence and blow-up phenomena for the Degasperis--Procesi equation. Comm. Math. Phys. 2006;267:801–820.
   Web of Science ®Google Scholar
• Yin Z. Global solutions to a new integrable equation with peakons. Indiana Univ. Math. J. 2004;53:1189–1209.
   Web of Science ®Google Scholar
• Escher J, Liu Y, Yin Z. Global weak solutions and blow-up structure for the Degasperis--Procesi equation. J. Funct. Anal. 2006;241:457–485.
   Web of Science ®Google Scholar
• Escher J, Liu Y, Yin Z. Shock waves and blow-up phenomena for the periodic Degasperis--Procesi equation. Indiana Univ. Math. J. 2007;56:87–117.
   Web of Science ®Google Scholar
• Liu Y, Yin Z. On the blow-up phenomena for the Degasperis--Procesi equation. Int. Math. Res. Not. 2007;22 pp. Art. ID rnm117
   Web of Science ®Google Scholar
• Yin Z. On the Cauchy problem for an integrable equation with peakon solutions. Illinois J. Math. 2003;47:649–666.
   Web of Science ®Google Scholar
• Yin Z. Global weak solutions for a new periodic integrable equation with peakon solutions. J. Funct. Anal. 2004;212:182–194.
   Web of Science ®Google Scholar
• Coclite GM, Karlsen KH. On the well-posedness of the Degasperis--Procesi equation. J. Funct. Anal. 2006;233:60–91.
   Web of Science ®Google Scholar
• Lundmark H. Formation and dynamics of shock waves in the Degasperis--Procesi equation. J. Nonlinear Sci. 2007;17:169–198.
   Web of Science ®Google Scholar
• Hone ANW, Wang J. Integrable peakon equations with cubic nonlinearity. J. Phys. A. 2008;41:372002 10 pp.
   Google Scholar
• Wu X, Yin Z. Well-posedness and global existence for the Novikov equation. Ann. Sc. Norm. Super. Pisa Cl. Sci (5). 2012;11:707–727.
   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
• Yan W, Li Y, Zhang Y. The Cauchy problem for the integrable Novikov equation. J. Differ. Equ. 2012;253:298–318.
   Web of Science ®Google Scholar
• Yan W, Li Y, Zhang Y. The Cauchy problem for the Novikov equation. NoDEA Nonlinear Differ. Equ. Appl. 2013;20:1157–1169.
   Web of Science ®Google Scholar
• Lai S. Global weak solutions to the Novikov equation. J. Funct. Anal. 2013;265:520–544.
   Web of Science ®Google Scholar
• Wu X, Yin Z. Global weak solutions for the Novikov equation. J. Phys. A. 2011;44:055202. 17 pp.
   Google Scholar
• Bahouri H, Chemin J-Y, Danchin R. Fourier analysis and nonlinear partial differential equations. Berlin Heidelberg: Springer-Verlag; 2011.
   Google Scholar
• Danchin R. Fourier Analysis Methods for PDEs. Vol. 14, Lecture Notes. 2005.
   Google Scholar
• Constantin A. A note on the uniqueness of solutions of ordinary differential equations. Appl. Anal. 1997;64:273–276.
   Google Scholar
• Luo W, Yin Z. Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space. Nonlinear Anal. 2015;122:1–22.
   Web of Science ®Google Scholar
• Li J, Yin Z. Well-posedness and global existence for a generalized Degasperis--Procesi equation. Nonlinear Anal. Real World Appl. 2016;28:72–90.
   Web of Science ®Google Scholar
• Zhang Z, Yin Z. Well-posedness, global existence and blow-up phenomena for an integrable multi-component Camassa--Holm system. Available from: http://arxiv.org/abs/1411.6402. Preprint arXiv: /1411.6402.pdf.
   Google Scholar