# American Institute of Mathematical Sciences

May  2014, 13(3): 1283-1304. doi: 10.3934/cpaa.2014.13.1283

## On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity

 1 Department of mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China 2 Department of Mathematics, University of Texas-Pan American, Edinburg, Texas 78541, United States 3 Department of Mathematics, Zhongshan University, 510275 Guangzhou, China

Received  June 2013 Revised  October 2013 Published  December 2013

In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly establish the local well-posedness result in Besov spaces, and then present a precise blow-up scenario for strong solutions. Furthermore, we show the existence of single peakon by the method of analysis.
Citation: Xingxing Liu, Zhijun Qiao, Zhaoyang Yin. On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1283-1304. doi: 10.3934/cpaa.2014.13.1283
##### References:
 [1] H. Bahouri, Y. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Pratial Differential Equations,, Springer-Verlag, (2011). doi: 10.1007/978-3-642-16830-7. [2] P. M. Bies, P. Górka and E. Reyes, The dual modified Korteweg-de Vries-Fokas-Qiao equation: Geometry and local analysis,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.4736845. [3] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, \emph{Phys. Lett.}, 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. [4] R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, \emph{Adv. Appl. Mech.}, 31 (1994), 1. [5] A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: a geometric approach,, \emph{Ann. Inst. Fourier(Grenoble)}, 50 (2000), 321. [6] A. Constantin, The trajectories of particles in Stokes waves,, \emph{Invent. Math.}, 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. [7] A. Constantin, Particle trajectories in extreme Stokes waves,, \emph{IMA J. Appl. Math.}, 77 (2012), 293. doi: 10.1093/imamat/hxs033. [8] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, \emph{Acta Math.}, 181 (1998), 229. doi: 10.1007/BF02392586. [9] A. Constantin and J. Escher, Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation,, \emph{Comm. Pure Appl. Math.}, 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [10] A. Constantin and J. Escher, Particle trajectories in solitary water waves,, \emph{Bull. Amer. Math. Soc.}, 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7. [11] A. Constantin and W. Strauss, Stability of peakons,, \emph{Comm. Pure Appl. Math.}, 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C. [12] A. Constantin and W. Strauss, Stability of a class of solitary waves in compressible elastic rods,, \emph{Phys. Lett. A}, 270 (2000), 140. doi: 10.1016/S0375-9601(00)00255-3. [13] A. Constantin and W. Strauss, Stability of the Camassa-Holm solitons,, \emph{J. Nonlinear. Sci.}, 12 (2002), 415. doi: 10.1007/s00332-002-0517-x. [14] R. Danchin, A few remarks on the Camassa-Holm equation,, \emph{Diff. Int. Eq.}, 14 (2001), 953. [15] A. Fokas, The Korteweg-de Vries equation and beyond,, \emph{Acta Appl. Math.}, 39 (1995), 295. doi: 10.1007/BF00994638. [16] A. Fokas, On a class of physically important integrable equations,, \emph{Physica D}, 87 (1995), 145. doi: 10.1016/0167-2789(95)00133-O. [17] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries,, \emph{Physica D}, 4 (1981), 47. doi: 10.1016/0167-2789(81)90004-X. [18] Y. Fu, G. Gui, C. Qu and Y. Liu, On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity,, \emph{J. Differential Equations}, 255 (2013), 1905. doi: 10.1016/j.jde.2013.05.024. [19] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation,, \emph{Physica D}, 95 (1996), 229. doi: 10.1016/0167-2789(96)00048-6. [20] G. Gui, Y. Liu, P. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation,, \emph{Comm. Math. Phys.}, 319 (2013), 731. doi: 10.1007/s00220-012-1566-0. [21] A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow water equation,, \emph{Diff. Int. Eq.}, 14 (2001), 821. [22] R. Ivanov and T. Lyons, Dark solitons of the Qiao's hierarchy,, \emph{J. Math. Phys.}, 53 (2012). [23] J. B. Li and Y. Zhang, Exact M/W-shape solitary wave solutions determined by a singular traveling wave equation,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 1797. doi: 10.1016/j.nonrwa.2008.02.016. [24] Y. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation,, \emph{J. Differential Equations}, 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. [25] X. Liu and Z. Yin, Local well-posedness and stability of peakons for a generalized Dullin-Gottwald-Holm equation,, \emph{Nonlinear Anal. TMA}, 74 (2011), 2497. doi: 10.1016/j.na.2010.12.005. [26] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, \emph{Phys. Rev. E}, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900. [27] Z. Qiao, A new integrable equation with cuspon and W/M-shape-peaks solitons,, \emph{J. Math. Phys.}, 47 (2006). doi: 10.1063/1.2365758. [28] Z. Qiao and X. Li, An integrable equation with nonsmooth solitons,, \emph{Theor. Math. Phys.}, 267 (2011), 584. [29] Z. Qiao, B. Xia and J. B. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions,, preprint, (). [30] C. Qu, X. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity,, \emph{Comm. Math. Phys.}, 322 (2013), 967. doi: 10.1007/s00220-013-1749-3. [31] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, \emph{Nonlinear Anal. TMA}, 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X. [32] S. Sakovich, Smooth soliton solutions of a new integrable equation by Qiao,, \emph{J. Math. Phys.}, 52 (2011). doi: 10.1063/1.3548837. [33] J. F. Toland, Stokes waves,, \emph{Topol. Methods Nonlinear Anal.}, 7 (1996), 1. [34] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions,, \emph{Illinois. J. Math.}, 47 (2003), 649.

show all references

##### References:
 [1] H. Bahouri, Y. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Pratial Differential Equations,, Springer-Verlag, (2011). doi: 10.1007/978-3-642-16830-7. [2] P. M. Bies, P. Górka and E. Reyes, The dual modified Korteweg-de Vries-Fokas-Qiao equation: Geometry and local analysis,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.4736845. [3] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, \emph{Phys. Lett.}, 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. [4] R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, \emph{Adv. Appl. Mech.}, 31 (1994), 1. [5] A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: a geometric approach,, \emph{Ann. Inst. Fourier(Grenoble)}, 50 (2000), 321. [6] A. Constantin, The trajectories of particles in Stokes waves,, \emph{Invent. Math.}, 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. [7] A. Constantin, Particle trajectories in extreme Stokes waves,, \emph{IMA J. Appl. Math.}, 77 (2012), 293. doi: 10.1093/imamat/hxs033. [8] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, \emph{Acta Math.}, 181 (1998), 229. doi: 10.1007/BF02392586. [9] A. Constantin and J. Escher, Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation,, \emph{Comm. Pure Appl. Math.}, 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [10] A. Constantin and J. Escher, Particle trajectories in solitary water waves,, \emph{Bull. Amer. Math. Soc.}, 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7. [11] A. Constantin and W. Strauss, Stability of peakons,, \emph{Comm. Pure Appl. Math.}, 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C. [12] A. Constantin and W. Strauss, Stability of a class of solitary waves in compressible elastic rods,, \emph{Phys. Lett. A}, 270 (2000), 140. doi: 10.1016/S0375-9601(00)00255-3. [13] A. Constantin and W. Strauss, Stability of the Camassa-Holm solitons,, \emph{J. Nonlinear. Sci.}, 12 (2002), 415. doi: 10.1007/s00332-002-0517-x. [14] R. Danchin, A few remarks on the Camassa-Holm equation,, \emph{Diff. Int. Eq.}, 14 (2001), 953. [15] A. Fokas, The Korteweg-de Vries equation and beyond,, \emph{Acta Appl. Math.}, 39 (1995), 295. doi: 10.1007/BF00994638. [16] A. Fokas, On a class of physically important integrable equations,, \emph{Physica D}, 87 (1995), 145. doi: 10.1016/0167-2789(95)00133-O. [17] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries,, \emph{Physica D}, 4 (1981), 47. doi: 10.1016/0167-2789(81)90004-X. [18] Y. Fu, G. Gui, C. Qu and Y. Liu, On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity,, \emph{J. Differential Equations}, 255 (2013), 1905. doi: 10.1016/j.jde.2013.05.024. [19] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation,, \emph{Physica D}, 95 (1996), 229. doi: 10.1016/0167-2789(96)00048-6. [20] G. Gui, Y. Liu, P. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation,, \emph{Comm. Math. Phys.}, 319 (2013), 731. doi: 10.1007/s00220-012-1566-0. [21] A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow water equation,, \emph{Diff. Int. Eq.}, 14 (2001), 821. [22] R. Ivanov and T. Lyons, Dark solitons of the Qiao's hierarchy,, \emph{J. Math. Phys.}, 53 (2012). [23] J. B. Li and Y. Zhang, Exact M/W-shape solitary wave solutions determined by a singular traveling wave equation,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 1797. doi: 10.1016/j.nonrwa.2008.02.016. [24] Y. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation,, \emph{J. Differential Equations}, 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. [25] X. Liu and Z. Yin, Local well-posedness and stability of peakons for a generalized Dullin-Gottwald-Holm equation,, \emph{Nonlinear Anal. TMA}, 74 (2011), 2497. doi: 10.1016/j.na.2010.12.005. [26] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, \emph{Phys. Rev. E}, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900. [27] Z. Qiao, A new integrable equation with cuspon and W/M-shape-peaks solitons,, \emph{J. Math. Phys.}, 47 (2006). doi: 10.1063/1.2365758. [28] Z. Qiao and X. Li, An integrable equation with nonsmooth solitons,, \emph{Theor. Math. Phys.}, 267 (2011), 584. [29] Z. Qiao, B. Xia and J. B. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions,, preprint, (). [30] C. Qu, X. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity,, \emph{Comm. Math. Phys.}, 322 (2013), 967. doi: 10.1007/s00220-013-1749-3. [31] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, \emph{Nonlinear Anal. TMA}, 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X. [32] S. Sakovich, Smooth soliton solutions of a new integrable equation by Qiao,, \emph{J. Math. Phys.}, 52 (2011). doi: 10.1063/1.3548837. [33] J. F. Toland, Stokes waves,, \emph{Topol. Methods Nonlinear Anal.}, 7 (1996), 1. [34] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions,, \emph{Illinois. J. Math.}, 47 (2003), 649.
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