References
- Alvarez-Samaniego B, Lannes D. Large time existence for 3D water-waves and asymptotics. Invent. Math. 2008;171:485–541.
- McKean HP. Integrable systems and algebraic curves, global analysis. Springer Lect. Notes Math. 1979;755:83–200.
- Tiglay F. The periodic cauchy problem for Novikov’s equation. Int. Math. Res. Not. 2011;20:4633–4648.
- Escauriaza L, Kenig CE, Ponce G, Vega L. On uniqueness properties of solutions of the k-generalized KdV equations. J. Funct. Anal. 2007;244:504–535.
- Kenig C, Ponce G, Vega L. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 1993;46:527–620.
- Whitham GB. Linear and nonlinear waves. New York: J. Wiley; 1980.
- Fokas A, Fuchssteiner B. Symplectic structures, their Bäcklund transformation and hereditary symmetries. Phys. D. 1981;4:47–66.
- Camassa R, Holm D. An integrable shallow water equation with peaked solitons. Phys. Rev. Letters. 1993;71:1661–1664.
- Lenells J. Conservation laws of the Camassa-Holm equation. J. Phys. A. 2005;38:869–880.
- Constantin A. On the scattering problem for the Camassa-Holm equation. Proc. Roy. Soc. London. 2001;457:953–970.
- Ivanov R. Extended Camassa-Holm hierarchy and conserved quantities. Z. Naturforsch. A. 2006;61:133–138.
- Boutet de Monvel A, Shepelsky D. Riemann-Hilbert approach for the Camassa-Holm equation on the line. C. R. Math. Acad. Sci. Paris. 2006;343:627–632.
- Constantin A, Gerdjikov V, Ivanov R. Inverse scattering transform for the Camassa-Holm equation. Inverse Problems. 2006;22:2197–2207.
- Constantin A, Strauss WA. Stability of the Camassa-Holm solitons. J. Nonlinear. Sci. 2002;12:415–422.
- Constantin A, Strauss WA. Stability of peakons. Comm. Pure Appl. Math. 2000;53:603–610.
- Constantin A, Escher J. Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 2011;173:559–568.
- Beals R, Sattinger D, Szmigielski J. Acoustic scattering and the extended Korteweg-de Vries hierarchy. Adv. Math. 1998;140:190–206.
- Constantin A, Escher J. Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Super. Pisa-Cl. Sci. 1998;26:303–328.
- Li YA, Olver PJ. Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Equ. 2000;162:27–63.
- Danchin R. A few remarks on the Camassa-Holm equation. Differ. Int. Equ. 2001;14:953–988.
- Constantin A. Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble). 2000;50:321–362.
- Constantin A, Escher J. Wave breaking for nonlinear nonlocal shallow water equations. Acta Mathematica. 1998;181:229–243.
- Constantin A, Ivanov R. On an integrable two-component Camassa-Holm shallow water system. Phys. Lett. A. 2008;372:7129–7132.
- Bressan A, Constantin A. Global conservative solutions of the Camassa-Holm equation. Arch. Rat. Mech. Anal. 2007;183:215–239.
- Constantin A, Escher J. Global weak solutions for a shallow water equation. Indiana. Univ. Math. J. 1998;47:1527–1545.
- Constantin A, Molinet L. Global weak solutions for a shallow water equation. Comm. Math. Phys. 2000;211:45–61.
- Xin ZP, Zhang P. On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. 2000;53:1411–1433.
- Camassa R, Holm D, Hyman J. A new integrable shallow water equation. Adv. Appl. Mech. 1994;31:1–33.
- Hakkaev S, Kirchev K. Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation. Comm. Partial Differ. Equ. 2005;30:761–781.
- Hakkaev S, Kirchev K. On the well-posedness and stability of peakons for a generalized Camassa-Holm equation. Int. J. Nonl. Sci. 2006;1:139–148.
- Mi YS, Mu CL. Well-posedness and analyticity for the Cauchy problem for the generalized Camassa-Holm equation. J. Math. Anal. Appl. 2013;405:173–182.
- Dullin HR, Gottwald GA, Holm DD. An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 2001;87:1945–1948.
- Ai X, Gui G. On the inverse scattering problem and the low regularity solutions for the Dullin-Gottwald-Holm equation. Nonl. Anal. RWA. 2010;11:888–894.
- Guo Z, Ni L. Wave breaking for the periodic weakly dissipative Dullin-Gottwald-Holm equation. Nonl. Anal. TMA. 2011;74:965–973.
- Meng Q, He B, Long Y, Li Z. New exact periodic wave solutions for the Dullin-Gottwald-Holm equation. Appl. Math. Comput. 2011;218:4533–4537.
- Shen C, Tian L, Gao A. Optimal control of the viscous Dullin-Gottwalld-Holm equation. Nonl. Anal. RWA. 2010;11:480–491.
- Sun B. Maximum principle for optimal distributed control of the viscous Dullin-Gottwald-Holm equation. Nonl. Anal. RWA. 2012;13:325–332.
- Liu Y. Global existence and blow-up solutions for a nonlinear shallow water equation. Math. Ann. 2006;335:717–735.
- Tian L, Gui G, Liu Y. On the Cauchy problem and the scattering problem for the Dullin-Gottwald-Holm equation. Comm. Math. Phys. 2005;257:667–701.
- Yin ZY. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete Contin. Dyn. Syst. 2004;11:393–411.
- Yan K, Yan Z. On the solutions of the Dullin-Gottwald-Holm equation in Besov spaces. Nonl. Anal. RWA. 2012;13:2580–2592.
- Yin Z. Global existence and blow-up for a periodic integrable shallow water equation with linear and nonlinear dispersion. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 2005;12:87–101.
- Zhang S, Yin Z. On the blow-up phenomena of the periodic Dullin-Gottwald-Holm equation. J. Math. Phys. 2008;49:1–16.
- Zhang S, Yin Z. Global weak solutions for the Dullin-Gottwald-Holm equation. Nonl. Anal. TMA. 2010;72:1690–1700.
- Mustafa OG. Global conservative solutions of the Dullin-Gottwald-Holm equation. Discrete Contin. Dyn. Syst. 2007;19:575–594.
- Yan K, Yin Z. Analytic solutions of the Cauchy problem for two-component shallow water systems. Math. Z. 2011;269:1113–1127.
- Fokas AS. The Korteweg-de Vries equation and beyond. Acta Appl. Math. 1995;39:295–305.
- Fuchssteiner B. Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation. Physica D. 1996;95:229–243.
- Olver PJ, Rosenau P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E. 1996;53:1900–1906.
- Qiao Z. A new integrable equation with cuspons and W/M-shape-peaks solitons. J. Math. Phys. 2006; 47.112701.9p .
- Qiao Z. New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solitons. J. Math. Phys. 2007; 48.082701.20p.
- Ivanov RI, Lyons T. Dark solitons of thr Qiao’s hierarchy. J. Math. Phys. 2012; 53.123701.8p.
- Gui G, Liu Y, Olver PJ, Qu C. Wave-breaking and peakons for a modified Camassa-Holm equation. Commu. Math. Phy. 2012;319:731–759.
- Fu Y, Gui G, Liu Y, Qu C. On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity. J. Differ. Equ. 2013;255:1905–1938.
- Schäfer-Wayne T. Propagation of ultra-short optical pulses in cubic nonlinear media. Physica D. 2004;196:90–105.
- Novikov VS. Generalizations of the Camassa-Holm equation. J. Phys. A. 2009;42. 342002. 14p.
- Mikhailov AV, Novikov VS. Perturbative symmetry approach. J. Phys. A. 2002;35:4775–4790.
- Hone ANW, Wang JP. Integrable peakon equations with cubic nonlinearity. J. Phys. Appl. Math. Theor. 2008; 41.372002.10p.
- Hone ANW, Lundmark H, Szmigielski J. Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm equation. Dyn. Partial Differ. Equ. 2009;6:253–289.
- Ni LD, Zhou Y. Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ. 2011;250:3002–3201.
- Wu XL, Yin ZY. A note on the Cauchy problem of the Novikov equation. Appl. Anal. 2013;92:1116–1137.
- Yan W, Li Y, Zhang Y. The Cauchy problem for the integrable Novikov equation. J. Differ. Equ. 2012;253:298–318.
- Yan W, Li YS, Zhang YM. The cauchy problem for the Novikov equation. Nonlinear Differ. Equ. Appl. 2013;20:1157–1169.
- Jiang ZH, Ni LD. Blow-up phemomena for the integrable Novikov equation. J. Math. Appl. Anal. 2012;385:551–558.
- Wu SY, Yin ZY. Global weak solutions for the Novikov equation. J. Phys. A: Math. Theor. 2011; 44.055202.11p.
- Lai SY, Li N, Wu YH. The existence of global strong and weak solutions for the Novikov equation. J. Math. Anal. Appl. 2013;399:682–691.
- Zhou SM, Mu CL. The Cauchy problem for a generalized b-equation with higher-order nonlinearities in critical Besov spaces and weighted Lp spaces. Discrete Contin. Dyn. Syst. Ser. A. Accepted.
- Zhou SM, Mu CL, Wang LC. Well-posedness, blow-up phenomena and global existence for the generalized b-equation with higher-order nonlinearities and weak dissipation. Discrete Contin. Dyn. Syst. Ser. A. 2014;32:843–867.
- Zhou SM, Mu CL. The properties of solutions for a generalized b-family equation with higher-order nonlinearities and peakons. J. Nonlinear Sci. 2013;23:863–889.
- Li N, Lai SY, Li S, Wu M. The local and global existence of solutions for a generalized Camasa-Holm equation. Abstr. Appl. Anal. 2012;532369. 26p.
- Liu XX, Yin ZY. Local well-posedness and stability of peakons for a generalized Dullin-Gottwald-Holm equation. Nonl. Anal. 2011;74:2497–2507.
- Lai S, Wu Y. The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation. J. Differ. Equ. 2010;248:2038–2063.
- Zhou SM. Persistence properties for a generalized Camassa-Holm equation in weighted Lp spaces. J. Math. Anal. Appl. 2014;410:932–938.
- Danchin R. Fourier analysis methods for PDEs. Lect. Notes. 2005;14:1–91.
- Danchin R. A note on well-posedness for Camassa-Holm equation. J. Differ. Equ. 2003;192:429–444.
- Tao T. Low-regularity global solutions to nonlinear dispersive equations, Surveys in analysis and operator theory (Canberra 2001). Proc. Centre Math. Appl. Austral. Nat. Univ. 40, Austral. Nat. Univ. Canberra. 2002;19–48.