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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 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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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.

[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. 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.

[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.

[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. YangY. 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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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.

[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. 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.

[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.

[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. YangY. 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.

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