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.

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[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.

[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.

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