On the Cauchy problem for a generalized Camassa-Holm equationwith both quadratic and cubic nonlinearity

Xingxing Liu; Zhijun Qiao; Zhaoyang Yin

doi:10.3934/cpaa.2014.13.1283

Article Contents

2014, Volume 13, Issue 3: 1283-1304. Doi: 10.3934/cpaa.2014.13.1283

This issue Previous Article On the orbital stability of fractional Schrödinger equations Next Article Sobolev norm estimates for a class of bilinear multipliers

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
2.
Department of Mathematics, University of Texas-Pan American, Edinburg, Texas 78541
3.
Department of Mathematics, Zhongshan University, 510275 Guangzhou

Received: May 31, 2013

Revised: September 30, 2013

Published: April 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 35G25, 35L05; Secondary: 35B30.

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References

[1]	H. Bahouri, Y. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Pratial Differential Equations, Springer-Verlag, Berlin Heidelberg, 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, J. Math. Phys., 53 (2012), 073710.doi: 10.1063/1.4736845.
[3]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Lett., 71 (1993), 1661-1664.doi: 10.1103/PhysRevLett.71.1661.
[4]	R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.
[5]	A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier(Grenoble), 50 (2000), 321-362.
[6]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.doi: 10.1007/s00222-006-0002-5.
[7]	A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.doi: 10.1093/imamat/hxs033.
[8]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.doi: 10.1007/BF02392586.
[9]	A. Constantin and J. Escher, Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.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, Bull. Amer. Math. Soc., 44 (2007), 423-431.doi: 10.1090/S0273-0979-07-01159-7.
[11]	A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.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, Phys. Lett. A, 270 (2000), 140-148.doi: 10.1016/S0375-9601(00)00255-3.
[13]	A. Constantin and W. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear. Sci., 12 (2002), 415-422.doi: 10.1007/s00332-002-0517-x.
[14]	R. Danchin, A few remarks on the Camassa-Holm equation, Diff. Int. Eq., 14 (2001), 953-988.
[15]	A. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305.doi: 10.1007/BF00994638.
[16]	A. Fokas, On a class of physically important integrable equations, Physica D, 87 (1995), 145-150.doi: 10.1016/0167-2789(95)00133-O.
[17]	A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66.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, J. Differential Equations, 255 (2013), 1905-1938.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, Physica D, 95 (1996), 229-243.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, Comm. Math. Phys., 319 (2013), 731-759.doi: 10.1007/s00220-012-1566-0.
[21]	A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow water equation, Diff. Int. Eq., 14 (2001), 821-831.
[22]	R. Ivanov and T. Lyons, Dark solitons of the Qiao's hierarchy, J. Math. Phys., 53 (2012), 123701.
[23]	J. B. Li and Y. Zhang, Exact M/W-shape solitary wave solutions determined by a singular traveling wave equation, Nonlinear Anal. Real World Appl., 10 (2009), 1797-1802.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, J. Differential Equations, 162 (2000), 27-63.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, Nonlinear Anal. TMA, 74 (2011), 2497-2507.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, Phys. Rev. E, 53 (1996), 1900-1906.doi: 10.1103/PhysRevE.53.1900.
[27]	Z. Qiao, A new integrable equation with cuspon and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701.doi: 10.1063/1.2365758.
[28]	Z. Qiao and X. Li, An integrable equation with nonsmooth solitons, Theor. Math. Phys., 267 (2011), 584-589.
[29]	Z. Qiao, B. Xia and J. B. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions, preprint, arXiv:1205.2028.
[30]	C. Qu, X. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity, Comm. Math. Phys., 322 (2013), 967-997.doi: 10.1007/s00220-013-1749-3.
[31]	G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal. TMA, 46 (2001), 309-327.doi: 10.1016/S0362-546X(01)00791-X.
[32]	S. Sakovich, Smooth soliton solutions of a new integrable equation by Qiao, J. Math. Phys., 52 (2011), 023509.doi: 10.1063/1.3548837.
[33]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
[34]	Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois. J. Math., 47 (2003), 649-666.