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December  2021, 14(12): 4409-4437. doi: 10.3934/dcdss.2021123

Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation

Department of Mathematics, Research Institute of Basic Science, Incheon National University, Incheon 22012, Republic of Korea

* Corresponding author: Byungsoo Moon

Received  July 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

Fund Project: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2020R1F1A1A01048468)

This paper is devoted to studying the dynamical stability of periodic peaked solitary waves for the generalized modified Camassa-Holm equation. The equation is a generalization of the modified Camassa-Holm equation and it possesses the Hamiltonian structure shared by the modified Camassa-Holm equation. The equation admits the periodic peakons. It is shown that the periodic peakons are dynamically stable under small perturbations in the energy space.

Citation: Byungsoo Moon. Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4409-4437. doi: 10.3934/dcdss.2021123
References:
[1]

S. C. Anco and E. Recio, A general family of multi-peakon equations and their properties, J. Phys. A: Math. Theor., 52 (2019), 125203.  doi: 10.1088/1751-8121/ab03dd.  Google Scholar

[2]

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

[3]

R. CamassaD. D. Holm and J. M. Hyman, A new integral shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.  doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar

[4]

A. Chen and X. Lu, Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. S, 40 (2020), 1703-1735.  doi: 10.3934/dcds.2020090.  Google Scholar

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R. ChenS. Pan and B. Zhang, Global conservative solutions for a modified periodic coupled Camassa-Holm system, Electron. Res. Arch., 29 (2021), 1691-1708.  doi: 10.3934/era.2020087.  Google Scholar

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

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A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.  Google Scholar

[8]

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

[9]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328.   Google Scholar

[10]

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

[11]

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

[12]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), R51-R79.  doi: 10.1088/0305-4470/35/32/201.  Google Scholar

[13]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.  doi: 10.1007/s00014-003-0785-6.  Google Scholar

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A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

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A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation, Phys. D, 157 (2001), 75-89.  doi: 10.1016/S0167-2789(01)00298-6.  Google Scholar

[16]

A. Constantin and W. A. Strauss, Stability of peakons, Commun. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.  Google Scholar

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H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207.  doi: 10.1007/BF01170373.  Google Scholar

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H. R. DullinG. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dyn. Res., 33 (2003), 73-95.  doi: 10.1016/S0169-5983(03)00046-7.  Google Scholar

[19]

H. R. DullinG. A. Gottwald and D. D. Holm, On asymptotically equivalent shallow water wave equations, Phys. D, 190 (2004), 1-14.  doi: 10.1016/j.physd.2003.11.004.  Google Scholar

[20]

K. El Dika and L. Molinet, Stability of multipeakons, Ann. Inst. H. Poincaŕe Anal. NonLin., 26 (2009), 1517-1532.  doi: 10.1016/j.anihpc.2009.02.002.  Google Scholar

[21]

A. S. Fokas, On a class of physically important integrable equations, Phys. D, 87 (1995), 145-150.  doi: 10.1016/0167-2789(95)00133-O.  Google Scholar

[22]

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

[23]

A. S. Fokas and B. Fuchssteiner, On the structure of symplectic operators and hereditary symmetries, Lett. Nuovo Cimento, 28 (1980), 299-303.  doi: 10.1007/BF02798794.  Google Scholar

[24]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Phys. D, 95 (1996), 229-243.  doi: 10.1016/0167-2789(96)00048-6.  Google Scholar

[25]

G. GuiY. LiuP. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Commun. Math. Phys., 319 (2013), 731-759.  doi: 10.1007/s00220-012-1566-0.  Google Scholar

[26]

Z. GuoX. LiuX. Liu and C. Qu, Stability of peakons for the generalized modified Camassa-Holm equation, J. Differential Equations, 266 (2019), 7749-7779.  doi: 10.1016/j.jde.2018.12.014.  Google Scholar

[27]

C. He and C. Qu, Global weak solutions for the two-component Novikov equation, Electron. Res. Arch., 28 (2020), 1545-1562.  doi: 10.3934/era.2020081.  Google Scholar

[28]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479.  doi: 10.1088/0951-7715/25/2/449.  Google Scholar

[29]

D. Holm and R. Ivanov, Smooth and peaked solitons of the Camassa-Holm equation and applications, J. Geom. Symm. Phys., 22 (2011), 13-49.  doi: 10.7546/jgsp-22-2011-13-49.  Google Scholar

[30]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.  Google Scholar

[31]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A: Math. Theor., 41 (2008), 372002.  doi: 10.1088/1751-8113/41/37/372002.  Google Scholar

[32]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224.  Google Scholar

[33]

R. S. Johnson, On solutions of the Camassa-Holm equation, Proc. R. Soc. Lond. A, 459 (2003), 1687-1708.  doi: 10.1098/rspa.2002.1078.  Google Scholar

[34]

R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dyn. Res., 33 (2003), 97-111.  doi: 10.1016/S0169-5983(03)00036-4.  Google Scholar

[35]

J. Lenells, Stability of periodic peakons, Int. Math. Res. Not., 2004 (2004), 485-499.  doi: 10.1155/S1073792804132431.  Google Scholar

[36]

J. Lenells, A variational approach to the stability of periodic peakons, J. Nonlin. Math. Phys., 11 (2004), 151-163.  doi: 10.2991/jnmp.2004.11.2.2.  Google Scholar

[37]

J. LenellsG. Misiolek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Commun. Math. Phys., 299 (2010), 129-161.  doi: 10.1007/s00220-010-1069-9.  Google Scholar

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Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[39]

X. Liu, Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity, Discrete Contin. Dyn. Syst. A, 38 (2018), 5505-5521.  doi: 10.3934/dcds.2018242.  Google Scholar

[40]

X. LiuY. Liu and C. Qu, Orbital stability of 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

[41]

X. LiuY. Liu and C. Qu, Stability of peakons for the Novikov equation, J. Math. Pures Appl., 101 (2014), 172-187.  doi: 10.1016/j.matpur.2013.05.007.  Google Scholar

[42]

Y. LiuC. Qu and Y. Zhang, Stability of peridic peakons for the modified $\mu$-Camassa-Holm equation, Phys. D, 250 (2013), 66-74.  doi: 10.1016/j.physd.2013.02.001.  Google Scholar

[43]

Y. Matsuno, The peakon limit of the N-soliton solution of the Camassa-Holm equation, J. Phys. Soc. Jpn., 76 (2007), 034003.  doi: 10.1143/JPSJ.76.034003.  Google Scholar

[44]

G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Birasoro group, J. Geom. Phys., 24 (1998), 203-208.  doi: 10.1016/S0393-0440(97)00010-7.  Google Scholar

[45]

B. Moon, Single peaked traveling wave solutions to a generalized $\mu$-Novikov equation, Adv. Nonlinear Anal., 10 (2021), 66-75.  doi: 10.1515/anona-2020-0106.  Google Scholar

[46]

V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42 (2009), 342002.  doi: 10.1088/1751-8113/42/34/342002.  Google Scholar

[47]

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

[48]

P. Popivanov and A. Slavova, Nonlinear Waves: An Introduction, World Scientific, Hackensack, (2011).  doi: 10.1142/7867.  Google Scholar

[49]

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

[50]

Z. Qiao and X. Li, An integrable equation with nonsmooth solitons, Theor. Math. Phys., 167 (2011), 584-589.  doi: 10.1007/s11232-011-0044-8.  Google Scholar

[51]

C. QuY. Fu and Y. Liu, Well-posedness, wave breaking and peakons for a modified $\mu$-Camassa-Holm equation, J. Funct. Anal., 266 (2014), 433-477.  doi: 10.1016/j.jfa.2013.09.021.  Google Scholar

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

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

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X. Wu and Z. Yin, Global weak solutions for the Novikov equation, J. Phys. A: Math. Theor., 44 (2011), 055202.  doi: 10.1088/1751-8113/44/5/055202.  Google Scholar

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R. Xu and Y. Yang, Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect, Disctete Contin. Dyn. Syst., 40 (2020), 6507-6527.  doi: 10.3934/dcds.2020288.  Google Scholar

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show all references

References:
[1]

S. C. Anco and E. Recio, A general family of multi-peakon equations and their properties, J. Phys. A: Math. Theor., 52 (2019), 125203.  doi: 10.1088/1751-8121/ab03dd.  Google Scholar

[2]

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

[3]

R. CamassaD. D. Holm and J. M. Hyman, A new integral shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.  doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar

[4]

A. Chen and X. Lu, Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. S, 40 (2020), 1703-1735.  doi: 10.3934/dcds.2020090.  Google Scholar

[5]

R. ChenS. Pan and B. Zhang, Global conservative solutions for a modified periodic coupled Camassa-Holm system, Electron. Res. Arch., 29 (2021), 1691-1708.  doi: 10.3934/era.2020087.  Google Scholar

[6]

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

[7]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.  Google Scholar

[8]

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

[9]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328.   Google Scholar

[10]

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

[11]

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

[12]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), R51-R79.  doi: 10.1088/0305-4470/35/32/201.  Google Scholar

[13]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.  doi: 10.1007/s00014-003-0785-6.  Google Scholar

[14]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[15]

A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation, Phys. D, 157 (2001), 75-89.  doi: 10.1016/S0167-2789(01)00298-6.  Google Scholar

[16]

A. Constantin and W. A. Strauss, Stability of peakons, Commun. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.  Google Scholar

[17]

H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207.  doi: 10.1007/BF01170373.  Google Scholar

[18]

H. R. DullinG. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dyn. Res., 33 (2003), 73-95.  doi: 10.1016/S0169-5983(03)00046-7.  Google Scholar

[19]

H. R. DullinG. A. Gottwald and D. D. Holm, On asymptotically equivalent shallow water wave equations, Phys. D, 190 (2004), 1-14.  doi: 10.1016/j.physd.2003.11.004.  Google Scholar

[20]

K. El Dika and L. Molinet, Stability of multipeakons, Ann. Inst. H. Poincaŕe Anal. NonLin., 26 (2009), 1517-1532.  doi: 10.1016/j.anihpc.2009.02.002.  Google Scholar

[21]

A. S. Fokas, On a class of physically important integrable equations, Phys. D, 87 (1995), 145-150.  doi: 10.1016/0167-2789(95)00133-O.  Google Scholar

[22]

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

[23]

A. S. Fokas and B. Fuchssteiner, On the structure of symplectic operators and hereditary symmetries, Lett. Nuovo Cimento, 28 (1980), 299-303.  doi: 10.1007/BF02798794.  Google Scholar

[24]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Phys. D, 95 (1996), 229-243.  doi: 10.1016/0167-2789(96)00048-6.  Google Scholar

[25]

G. GuiY. LiuP. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Commun. Math. Phys., 319 (2013), 731-759.  doi: 10.1007/s00220-012-1566-0.  Google Scholar

[26]

Z. GuoX. LiuX. Liu and C. Qu, Stability of peakons for the generalized modified Camassa-Holm equation, J. Differential Equations, 266 (2019), 7749-7779.  doi: 10.1016/j.jde.2018.12.014.  Google Scholar

[27]

C. He and C. Qu, Global weak solutions for the two-component Novikov equation, Electron. Res. Arch., 28 (2020), 1545-1562.  doi: 10.3934/era.2020081.  Google Scholar

[28]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479.  doi: 10.1088/0951-7715/25/2/449.  Google Scholar

[29]

D. Holm and R. Ivanov, Smooth and peaked solitons of the Camassa-Holm equation and applications, J. Geom. Symm. Phys., 22 (2011), 13-49.  doi: 10.7546/jgsp-22-2011-13-49.  Google Scholar

[30]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.  Google Scholar

[31]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A: Math. Theor., 41 (2008), 372002.  doi: 10.1088/1751-8113/41/37/372002.  Google Scholar

[32]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224.  Google Scholar

[33]

R. S. Johnson, On solutions of the Camassa-Holm equation, Proc. R. Soc. Lond. A, 459 (2003), 1687-1708.  doi: 10.1098/rspa.2002.1078.  Google Scholar

[34]

R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dyn. Res., 33 (2003), 97-111.  doi: 10.1016/S0169-5983(03)00036-4.  Google Scholar

[35]

J. Lenells, Stability of periodic peakons, Int. Math. Res. Not., 2004 (2004), 485-499.  doi: 10.1155/S1073792804132431.  Google Scholar

[36]

J. Lenells, A variational approach to the stability of periodic peakons, J. Nonlin. Math. Phys., 11 (2004), 151-163.  doi: 10.2991/jnmp.2004.11.2.2.  Google Scholar

[37]

J. LenellsG. Misiolek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Commun. Math. Phys., 299 (2010), 129-161.  doi: 10.1007/s00220-010-1069-9.  Google Scholar

[38]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[39]

X. Liu, Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity, Discrete Contin. Dyn. Syst. A, 38 (2018), 5505-5521.  doi: 10.3934/dcds.2018242.  Google Scholar

[40]

X. LiuY. Liu and C. Qu, Orbital stability of 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

[41]

X. LiuY. Liu and C. Qu, Stability of peakons for the Novikov equation, J. Math. Pures Appl., 101 (2014), 172-187.  doi: 10.1016/j.matpur.2013.05.007.  Google Scholar

[42]

Y. LiuC. Qu and Y. Zhang, Stability of peridic peakons for the modified $\mu$-Camassa-Holm equation, Phys. D, 250 (2013), 66-74.  doi: 10.1016/j.physd.2013.02.001.  Google Scholar

[43]

Y. Matsuno, The peakon limit of the N-soliton solution of the Camassa-Holm equation, J. Phys. Soc. Jpn., 76 (2007), 034003.  doi: 10.1143/JPSJ.76.034003.  Google Scholar

[44]

G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Birasoro group, J. Geom. Phys., 24 (1998), 203-208.  doi: 10.1016/S0393-0440(97)00010-7.  Google Scholar

[45]

B. Moon, Single peaked traveling wave solutions to a generalized $\mu$-Novikov equation, Adv. Nonlinear Anal., 10 (2021), 66-75.  doi: 10.1515/anona-2020-0106.  Google Scholar

[46]

V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42 (2009), 342002.  doi: 10.1088/1751-8113/42/34/342002.  Google Scholar

[47]

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

[48]

P. Popivanov and A. Slavova, Nonlinear Waves: An Introduction, World Scientific, Hackensack, (2011).  doi: 10.1142/7867.  Google Scholar

[49]

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

[50]

Z. Qiao and X. Li, An integrable equation with nonsmooth solitons, Theor. Math. Phys., 167 (2011), 584-589.  doi: 10.1007/s11232-011-0044-8.  Google Scholar

[51]

C. QuY. Fu and Y. Liu, Well-posedness, wave breaking and peakons for a modified $\mu$-Camassa-Holm equation, J. Funct. Anal., 266 (2014), 433-477.  doi: 10.1016/j.jfa.2013.09.021.  Google Scholar

[52]

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

[53]

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

[54]

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