Abstract
This paper deals with the Cauchy problem for a shallow water equation with high-order nonlinearities, y t +u m+1 y x +bu m u x y=0, where b is a constant, \(m\in \mathbb{N}\), and we have the notation \(y:= (1-\partial_{x}^{2}) u\) , which includes the famous Camassa–Holm equation, the Degasperis–Procesi equation, and the Novikov equation as special cases. The local well-posedness of strong solutions for the equation in each of the Sobolev spaces \(H^{s}(\mathbb{R})\) with \(s>\frac{3}{2}\) is obtained, and persistence properties of the strong solutions are studied. Furthermore, although the \(H^{1}(\mathbb{R})\)-norm of the solution to the nonlinear model does not remain constant, the existence of its weak solutions in each of the low order Sobolev spaces \(H^{s}(\mathbb{R})\) with \(1<s<\frac{3}{2}\) is established, under the assumption \(u_{0}(x)\in H^{s}(\mathbb{R})\cap W^{1,\infty}(\mathbb{R})\). Finally, the global weak solution and peakon solution for the equation are also given.
Similar content being viewed by others
References
Bona, J., Smith, R.: The initial value problem for the Korteweg–de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A 278, 555–601 (1975)
Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)
Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)
Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)
Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 26, 303–328 (1998)
Constantin, A., Escher, J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51, 475–504 (1998)
Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007)
Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)
Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)
Constantin, A., Strauss, W.A.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)
Degasperis, A., Procesi, M.: Asymptotic integrability. In: Degasperis, A., Gaeta, G. (eds.) Symmetry and Perturbation Theory, pp. 23–37. World Scientific, Singapore (1999)
Degasperis, A., Holm, D.D., Hone, A.N.W.: A new integrable equation with peakon solutions. Theor. Math. Phys. 133, 1461–1472 (2002)
Degasperis, A., Holm, D.D., Hone, A.N.W.: Integrable and non-integrable equations with peakons. In: Nonlinear Physics: Theory and Experiment, II, Gallipoli, 2002, pp. 37–43. World Scientific, Singapore (2003)
Deng, S.F., Guo, B.L., Wang, T.C.: Travelling wave solutions of a generalized Camassa–Holm–Degasperis–Procesi equation. Sci. China Math. 54, 55–572 (2011)
Dullin, H.R., Gottwald, G.A., Holm, D.D.: An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87, 4501–4504 (2001)
Dullin, H.R., Gottwald, G.A., Holm, D.D.: Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 33, 73–79 (2003)
Dullin, H.R., Gottwald, G.A., Holm, D.D.: On asymptotically equivalent shallow water wave equations. Phys. Rev. D 190, 1–14 (2004)
Escher, J., Yin, Z.: Well-posedness, blow-up phenomena, and global solutions for the b-equation. J. Reine Angew. Math. 624, 51–80 (2008)
Escher, J., Liu, Y., Yin, Z.: Global weak solutions and blow-up structure for the Degasperis–Procesi equation. J. Funct. Anal. 241, 457–485 (2006)
Gilson, C., Pickering, A.: Factorization and Painlevé analysis of a class of nonlinear third-order partial differential equations. J. Phys. A, Math. Gen. 28, 2871–2888 (1995)
Hakkaev, S., Kirchev, K.: Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa–Holm equation. Commun. Partial Differ. Equ. 30, 761–781 (2005)
Henry, D.: Persistence properties for a family of nonlinear partial differential equations. Nonlinear Anal. 70, 1565–1573 (2009)
Himonas, A.A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa–Holm equation. Commun. Math. Phys. 271, 511–522 (2007)
Holm, D.D., Staley, M.F.: Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dyn. Syst. 2, 323–380 (2003a)
Holm, D.D., Staley, M.F.: Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE. Phys. Lett. A 308, 437–444 (2003b)
Hone, A.N.W.: Painleve tests, singularity structure and integrability. Lect. Notes Phys. 767, 245–277 (2009)
Hone, A.N.W., Wang, J.P.: Integrable peakon equations with cubic nonlinearity. J. Phys. A 41, 372002 (2008) 10 pp.
Hone, A.N.W., Lundmark, H., Szmigielski, J.: Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa–Holm equation. Dyn. Partial Differ. Equ. 6, 253–289 (2009)
Ivanov, R.I.: Water waves and integrability. Philos. Trans. R. Soc. Lond. A 365, 2267–2280 (2007)
Jiang, Z.H., Ni, L.D.: Blow-up phenomena for the integrable Novikov equation. J. Math. Anal. Appl. 385, 551–558 (2012)
Kato, T.: Quasi-linear equations of evolution with applications to partial differential equations. In: Spectral Theory and Differential Equations. Lecture Notes in Math., vol. 448, pp. 25–70. Springer, Berlin (1975)
Lai, S.Y., Wu, Y.H.: Global solutions and blow-up phenomena to a shallow water equation. J. Differ. Equ. 249, 693–706 (2010)
Lai, S.Y., Wu, Y.H.: A model containing both the Camassa–Holm and Degasperis–Procesi equations. J. Math. Anal. Appl. 374, 458–469 (2011)
Li, Y.A., Olver, P.J.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Equ. 162, 27–63 (2000)
Li, N., Lai, S.Y., Li, S., Wu, W.: The local and global existence of solutions for a generalized Camasa-Holm equation. Abstr. Appl. Anal. 532369 (2012), 26 pp.
Liu, R.: Several new types of solitary wave solutions for the generalized Camassa–Holm–Degasperis–Procesi equation. Commun. Pure Appl. Anal. 9, 77–90 (2010)
Liu, Z.R., Guo, B.L.: Periodic blow-up solutions and their limit forms for the generalized Camassa–Holm equation. Prog. Nat. Sci. 18, 259–266 (2008)
Liu, Z.R., Ouyang, Z.Y.: A note on solitary waves for modified forms of Camassa–Holm and Degasperis–Procesi equations. Phys. Lett. A 366, 377–381 (2007)
Liu, Z.R., Qian, T.F.: Peakons and their bifurcation in a generalized Camassa–Holm equation. Int. J. Bifurc. Chaos 11, 781–792 (2001)
Liu, Y., Yin, Z.Y.: Global existence and blow-up phenomena for the Degasperis–Procesi equation. Commun. Math. Phys. 267, 801–820 (2006)
Mi, Y.S., Mu, C.L.: On the Cauchy problem for the modified Camassa-Holm equation with peakon solutions (2013, preprint)
Mikhailov, A.V., Novikov, V.S.: Perturbative symmetry approach. J. Phys. A 35, 4775–4790 (2002)
Mu, C.L., Zhou, S.M., Zeng, R.: Well-posedness and blow-up phenomena for a higher order shallow water equation. J. Differ. Equ. 251, 3488–3499 (2011)
Ni, L.D., Zhou, Y.: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ. 250, 3002–3201 (2011)
Ni, L.D., Zhou, Y.: A new asymptotic behavior of solutions to the Camassa–Holm equation. Proc. Am. Math. Soc. 140, 607–614 (2012)
Novikov, V.S.: Generalizations of the Camassa–Holm equation. J. Phys. A 42, 342002 (2009) 14 pp.
Shen, J.W., Xu, W.: Bifurcations of smooth and non-smooth traveling wave solutions in the generalized Camassa–Holm equation. Chaos Solitons Fractals 26, 1149–1162 (2005)
Tian, L.X., Song, X.Y.: New peaked solitary wave solutions of the generalized Camassa–Holm equation. Chaos Solitons Fractals 21, 621–637 (2004)
Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)
Walter, W.: Differential and Integral Inequalities. Springer, New York (1970)
Wazwaz, A.: Solitary wave solutions for modified forms of Degasperis–Procesi and Camassa–Holm equations. Phys. Lett. A 352, 500–504 (2006)
Wazwaz, A.: New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations. Appl. Math. Comput. 186, 130–141 (2007)
Wu, S.Y., Yin, Z.Y.: Global weak solutions for the Novikov equation. J. Phys. A 44, 055202 (2011) 17 pp.
Xin, Z.P., Zhang, P.: On the weak solutions to a shallow water equation. Commun. Pure Appl. Math. 53, 1411–1433 (2000)
Yan, W., Li, Y.S., Zhang, Y.M.: The Cauchy problem for the integrable Novikov equation. J. Differ. Equ. 253, 298–318 (2012)
Yin, Z.Y.: Global solutions to a new integrable equation with peakons. Indiana Univ. Math. J. 53, 1189–1210 (2004)
Zhang, L., Chen, L.Q., Huo, X.: Bifurcations of smooth and nonsmooth traveling wave solutions in a generalized Degasperis–Procesi equation. J. Comput. Appl. Math. 205, 174–185 (2007)
Acknowledgements
The authors are very grateful to the anonymous reviewers for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This work is supported in part by NSF of P.R. China (11071266) and in part by the found of Chongqing Normal University (13XLB006) and in part by Scholarship Award for Excellent Doctoral Student granted by Ministry of Education.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by D.D. Holm.
Rights and permissions
About this article
Cite this article
Zhou, S., Mu, C. The Properties of Solutions for a Generalized b-Family Equation with Peakons. J Nonlinear Sci 23, 863–889 (2013). https://doi.org/10.1007/s00332-013-9171-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-013-9171-8