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

Previous Article
Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms
DCDS-S Home
This Issue
Next Article
Inviscid limit for the damped generalized incompressible Navier-Stokes equations on $ \mathbb{T}^2 $

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

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

Byungsoo Moon ^,

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.

Keywords: Generalized modified Camassa-Holm equation, periodic peakon, orbital stability.

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

Citation: Byungsoo Moon. Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation. Discrete and 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.
[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.
[3]	R. Camassa, D. 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.
[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.
[5]	R. Chen, S. 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.
[6]	A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033.
[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.
[8]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.
[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.
[10]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	H. R. Dullin, G. 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.
[19]	H. R. Dullin, G. 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.
[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.
[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.
[22]	A. S. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305. doi: 10.1007/BF00994638.
[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.
[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.
[25]	G. Gui, Y. Liu, P. 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.
[26]	Z. Guo, X. Liu, X. 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.
[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.
[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.
[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.
[30]	D. D. Holm, J. 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.
[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.
[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.
[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.
[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.
[35]	J. Lenells, Stability of periodic peakons, Int. Math. Res. Not., 2004 (2004), 485-499. doi: 10.1155/S1073792804132431.
[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.
[37]	J. Lenells, G. 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.
[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.
[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.
[40]	X. Liu, Y. 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.
[41]	X. Liu, Y. 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.
[42]	Y. Liu, C. 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.
[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.
[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.
[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.
[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.
[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.
[48]	P. Popivanov and A. Slavova, Nonlinear Waves: An Introduction, World Scientific, Hackensack, (2011). doi: 10.1142/7867.
[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.
[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.
[51]	C. Qu, Y. 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.
[52]	C. Qu, X. 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.
[53]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001.
[54]	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.
[55]	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.
[56]	M. Yang, Y. Li and Y. Zhao, On the Cauchy problem of generalized Fokas-Olver-Resenau-Qiao equation, Appl. Anal., 97 (2018), 2246-2268. doi: 10.1080/00036811.2017.1359565.

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.
[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.
[3]	R. Camassa, D. 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.
[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.
[5]	R. Chen, S. 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.
[6]	A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033.
[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.
[8]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.
[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.
[10]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	H. R. Dullin, G. 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.
[19]	H. R. Dullin, G. 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.
[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.
[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.
[22]	A. S. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305. doi: 10.1007/BF00994638.
[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.
[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.
[25]	G. Gui, Y. Liu, P. 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.
[26]	Z. Guo, X. Liu, X. 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.
[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.
[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.
[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.
[30]	D. D. Holm, J. 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.
[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.
[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.
[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.
[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.
[35]	J. Lenells, Stability of periodic peakons, Int. Math. Res. Not., 2004 (2004), 485-499. doi: 10.1155/S1073792804132431.
[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.
[37]	J. Lenells, G. 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.
[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.
[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.
[40]	X. Liu, Y. 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.
[41]	X. Liu, Y. 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.
[42]	Y. Liu, C. 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.
[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.
[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.
[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.
[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.
[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.
[48]	P. Popivanov and A. Slavova, Nonlinear Waves: An Introduction, World Scientific, Hackensack, (2011). doi: 10.1142/7867.
[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.
[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.
[51]	C. Qu, Y. 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.
[52]	C. Qu, X. 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.
[53]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001.
[54]	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.
[55]	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.
[56]	M. Yang, Y. Li and Y. Zhao, On the Cauchy problem of generalized Fokas-Olver-Resenau-Qiao equation, Appl. Anal., 97 (2018), 2246-2268. doi: 10.1080/00036811.2017.1359565.

[1]	Aiyong Chen, Xinhui Lu. Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1703-1735. doi: 10.3934/dcds.2020090
[2]	Xingxing Liu. Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5505-5521. doi: 10.3934/dcds.2018242
[3]	Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027
[4]	Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781
[5]	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
[6]	Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067
[7]	Yongsheng Mi, Boling Guo, Chunlai Mu. Persistence properties for the generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1623-1630. doi: 10.3934/dcdsb.2019243
[8]	Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871
[9]	Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115
[10]	Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087
[11]	Guangying Lv, Mingxin Wang. Some remarks for a modified periodic Camassa-Holm system. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1161-1180. doi: 10.3934/dcds.2011.30.1161
[12]	Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194
[13]	Ying Fu. A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2011-2039. doi: 10.3934/dcds.2015.35.2011
[14]	Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure and Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501
[15]	Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230
[16]	Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713
[17]	Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure and Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45
[18]	Li Yang, Chunlai Mu, Shouming Zhou, Xinyu Tu. The global conservative solutions for the generalized camassa-holm equation. Electronic Research Archive, 2019, 27: 37-67. doi: 10.3934/era.2019009
[19]	Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459
[20]	Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065

2020 Impact Factor: 2.425

Tools

Metrics

Other articles
by authors

on AIMS
Byungsoo Moon
on Google Scholar
Byungsoo Moon

2020 Impact Factor: 2.425

Tools

Article outline

2020 Impact Factor: 2.425

Tools

Metrics

Other articles
by authors

on AIMS
Byungsoo Moon
on Google Scholar
Byungsoo Moon

[Back to Top]