November  2018, 38(11): 5505-5521. doi: 10.3934/dcds.2018242

Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity

Department of mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

Received  September 2017 Revised  November 2017 Published  August 2018

In this paper, we consider the orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity, which admits the single peakons and multi-peakons. We firstly show the existence of the single peakon and prove two useful conservation laws. Then by constructing certain Lyapunov functionals, we give the proof of stability result of peakons in the energy space $ H^1(\mathbb{R})$-norm.

Citation: Xingxing Liu. Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5505-5521. doi: 10.3934/dcds.2018242
References:
[1]

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, 19pp. doi: 10.1063/1.4736845.  Google Scholar

[2]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

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R. M. ChenY. LiuC. Qu and S. Zhang, Oscillation-induced blow-up to the modified Camassa-Holm equation with linear dispersion, Adv. Math., 272 (2015), 225-251.  doi: 10.1016/j.aim.2014.12.003.  Google Scholar

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

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A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[6]

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

[7]

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

[8]

Y. FuG. GuiC. 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

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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

[10]

G. GuiY. LiuP. 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

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A. Himonas and D. Mantzavinos, The Cauchy problem for the Fokas-Olver-Rosenau-Qiao equation, Nonlinear Anal. TMA, 95 (2014), 499-529.  doi: 10.1016/j.na.2013.09.028.  Google Scholar

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R. Ivanov and T. Lyons, Dark solitons of the Qiao's hierarchy, J. Math. Phys., 53 (2012), 123701, 8pp. doi: 10.1063/1.4764859.  Google Scholar

[13]

Z. W. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.  doi: 10.1002/cpa.20239.  Google Scholar

[14]

X. C. LiuY. Liu and C. Qu, Orbital stability of the train of peakons for an integrable modified Camassa-Holm equation, Adv. Math., 255 (2014), 1-37.  doi: 10.1016/j.aim.2013.12.032.  Google Scholar

[15]

Y. LiuP. J. OlverC. Qu and S. Zhang, On the blow-up solutions to the integrable modified Camassa-Holm equation, Anal. Appl., (Singap.), 12 (2014), 355-368.  doi: 10.1142/S0219530514500274.  Google Scholar

[16]

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

[17]

Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9pp. doi: 10.1063/1.2365758.  Google Scholar

[18]

Z. Qiao, New integrable hierarchy, parametric solutions, cuspons, one-peak solitons, and M/W-shape solutions, J. Math. Phys., 48 (2007), 082701, 20pp. doi: 10.1063/1.2759830.  Google Scholar

[19]

Z. Qiao and X. Li, An integrable equation with nonsmooth solitons, Theor. Math. Phys., 267 (2011), 584-589.  doi: 10.1007/s11232-011-0044-8.  Google Scholar

[20]

C. QuX. C. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonliearity, Comm. Math. Phys., 322 (2013), 967-997.  doi: 10.1007/s00220-013-1749-3.  Google Scholar

[21]

E. Recio and S. C. Anco, A general family of multi-peakon equations and their properties, https://arXiv.org/abs/1609.04354. Google Scholar

[22]

S. Sakovich, Smooth soliton solutions of a new integrable equation by Qiao, J. Math. Phys., 52 (2011), 023509, 9pp. doi: 10.1063/1.3548837.  Google Scholar

[23]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  doi: 10.12775/TMNA.1996.001.  Google Scholar

[24]

B. XiaZ. Qiao and J. B. Li, An integrable system with peakon, complex peakon, weak kink, and kink-peakon interactional solutions, Commun. Nonlinear Sci. Numer. Simul., 63 (2018), 292-306.  doi: 10.1016/j.cnsns.2018.03.019.  Google Scholar

[25]

M. Yang, Y. Li and Y. Zhao, On the Cauchy problem of generalized Fokas-Olver-Resenau -Qiao equation, Appl. Anal., https://doi.org/10.1080/00036811.2017.1359565 doi: 10.1080/00036811.2017.1359565.  Google Scholar

show all references

References:
[1]

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, 19pp. doi: 10.1063/1.4736845.  Google Scholar

[2]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[3]

R. M. ChenY. LiuC. Qu and S. Zhang, Oscillation-induced blow-up to the modified Camassa-Holm equation with linear dispersion, Adv. Math., 272 (2015), 225-251.  doi: 10.1016/j.aim.2014.12.003.  Google Scholar

[4]

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

[5]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[6]

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

[7]

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

[8]

Y. FuG. GuiC. 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

[9]

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

[10]

G. GuiY. LiuP. 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

[11]

A. Himonas and D. Mantzavinos, The Cauchy problem for the Fokas-Olver-Rosenau-Qiao equation, Nonlinear Anal. TMA, 95 (2014), 499-529.  doi: 10.1016/j.na.2013.09.028.  Google Scholar

[12]

R. Ivanov and T. Lyons, Dark solitons of the Qiao's hierarchy, J. Math. Phys., 53 (2012), 123701, 8pp. doi: 10.1063/1.4764859.  Google Scholar

[13]

Z. W. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.  doi: 10.1002/cpa.20239.  Google Scholar

[14]

X. C. LiuY. Liu and C. Qu, Orbital stability of the train of peakons for an integrable modified Camassa-Holm equation, Adv. Math., 255 (2014), 1-37.  doi: 10.1016/j.aim.2013.12.032.  Google Scholar

[15]

Y. LiuP. J. OlverC. Qu and S. Zhang, On the blow-up solutions to the integrable modified Camassa-Holm equation, Anal. Appl., (Singap.), 12 (2014), 355-368.  doi: 10.1142/S0219530514500274.  Google Scholar

[16]

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

[17]

Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9pp. doi: 10.1063/1.2365758.  Google Scholar

[18]

Z. Qiao, New integrable hierarchy, parametric solutions, cuspons, one-peak solitons, and M/W-shape solutions, J. Math. Phys., 48 (2007), 082701, 20pp. doi: 10.1063/1.2759830.  Google Scholar

[19]

Z. Qiao and X. Li, An integrable equation with nonsmooth solitons, Theor. Math. Phys., 267 (2011), 584-589.  doi: 10.1007/s11232-011-0044-8.  Google Scholar

[20]

C. QuX. C. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonliearity, Comm. Math. Phys., 322 (2013), 967-997.  doi: 10.1007/s00220-013-1749-3.  Google Scholar

[21]

E. Recio and S. C. Anco, A general family of multi-peakon equations and their properties, https://arXiv.org/abs/1609.04354. Google Scholar

[22]

S. Sakovich, Smooth soliton solutions of a new integrable equation by Qiao, J. Math. Phys., 52 (2011), 023509, 9pp. doi: 10.1063/1.3548837.  Google Scholar

[23]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  doi: 10.12775/TMNA.1996.001.  Google Scholar

[24]

B. XiaZ. Qiao and J. B. Li, An integrable system with peakon, complex peakon, weak kink, and kink-peakon interactional solutions, Commun. Nonlinear Sci. Numer. Simul., 63 (2018), 292-306.  doi: 10.1016/j.cnsns.2018.03.019.  Google Scholar

[25]

M. Yang, Y. Li and Y. Zhao, On the Cauchy problem of generalized Fokas-Olver-Resenau -Qiao equation, Appl. Anal., https://doi.org/10.1080/00036811.2017.1359565 doi: 10.1080/00036811.2017.1359565.  Google Scholar

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