American Institute of Mathematical Sciences

Advanced
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

Xingxing Liu 1, , Zhijun Qiao 2, and  Zhaoyang Yin 3,

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.
Keywords: local well-posedness, Besov spaces, blow-up scenario, Generalized Camassa-Holm equation, peakon..
Mathematics Subject Classification: Primary: 35G25, 35L05; Secondary: 35B3.
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, Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[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.  Google Scholar

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[7]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033.  Google Scholar

[8]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. Danchin, A few remarks on the Camassa-Holm equation, Diff. Int. Eq., 14 (2001), 953-988.  Google Scholar

[15]

A. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305. doi: 10.1007/BF00994638.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow water equation, Diff. Int. Eq., 14 (2001), 821-831.  Google Scholar

[22]

R. Ivanov and T. Lyons, Dark solitons of the Qiao's hierarchy, J. Math. Phys., 53 (2012), 123701. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

Z. Qiao and X. Li, An integrable equation with nonsmooth solitons, Theor. Math. Phys., 267 (2011), 584-589. Google Scholar

[29]

Z. Qiao, B. Xia and J. B. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

S. Sakovich, Smooth soliton solutions of a new integrable equation by Qiao, J. Math. Phys., 52 (2011), 023509. doi: 10.1063/1.3548837.  Google Scholar

[33]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  Google Scholar

[34]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois. J. Math., 47 (2003), 649-666.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[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.  Google Scholar

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[7]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033.  Google Scholar

[8]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. Danchin, A few remarks on the Camassa-Holm equation, Diff. Int. Eq., 14 (2001), 953-988.  Google Scholar

[15]

A. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305. doi: 10.1007/BF00994638.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow water equation, Diff. Int. Eq., 14 (2001), 821-831.  Google Scholar

[22]

R. Ivanov and T. Lyons, Dark solitons of the Qiao's hierarchy, J. Math. Phys., 53 (2012), 123701. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

Z. Qiao and X. Li, An integrable equation with nonsmooth solitons, Theor. Math. Phys., 267 (2011), 584-589. Google Scholar

[29]

Z. Qiao, B. Xia and J. B. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

S. Sakovich, Smooth soliton solutions of a new integrable equation by Qiao, J. Math. Phys., 52 (2011), 023509. doi: 10.1063/1.3548837.  Google Scholar

[33]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  Google Scholar

[34]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois. J. Math., 47 (2003), 649-666.  Google Scholar

[1]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781
[2]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure &amp; Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501
[3]

Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042
[4]

Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493
[5]

Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019
[6]

Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025
[7]

Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171
[8]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139
[9]

Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112
[10]

Wenxia Chen, Jingyi Liu, Danping Ding, Lixin Tian. Blow-up for two-component Camassa-Holm equation with generalized weak dissipation. Communications on Pure &amp; Applied Analysis, 2020, 19 (7) : 3769-3784. doi: 10.3934/cpaa.2020166
[11]

Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027
[12]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111
[13]

Kai Yan, Zhaoyang Yin. Well-posedness for a modified two-component Camassa-Holm system in critical spaces. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1699-1712. doi: 10.3934/dcds.2013.33.1699
[14]

Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115
[15]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[16]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure &amp; Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809
[17]

Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203
[18]

Stephen C. Anco, Elena Recio, María L. Gandarias, María S. Bruzón. A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Conference Publications, 2015, 2015 (special) : 29-37. doi: 10.3934/proc.2015.0029
[19]

Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021052
[20]

Yongsheng Mi, Boling Guo, Chunlai Mu. Persistence properties for the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1623-1630. doi: 10.3934/dcdsb.2019243

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences