• Previous Article
    Large time behavior of solution to quasilinear chemotaxis system with logistic source
  • DCDS Home
  • This Issue
  • Next Article
    On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay
March  2020, 40(3): 1703-1735. doi: 10.3934/dcds.2020090

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

1. 

Department of Mathematics, Hunan First Normal University, Changsha, Hunan 410205, China

2. 

School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China

Received  May 2019 Revised  September 2019 Published  December 2019

Fund Project: This work are supported by the National Natural Science Foundation of China (No. 11671107 and No. 11971163), Guangxi Natural Science Foundation of China (No. 2015GXNSFGA139004) and Program for Innovation Project of GUET Graduate Education (No. 2019YCXS080).

The orbital stability of peakons and hyperbolic periodic peakons for the Camassa-Holm equation has been established by Constantin and Strauss in [A. Constantin, W. Strauss, Comm. Pure. Appl. Math. 53 (2000) 603-610] and Lenells in [J. Lenells, Int. Math. Res. Not. 10 (2004) 485-499], respectively. In this paper, we prove the orbital stability of the elliptic periodic peakons for the modified Camassa-Holm equation. By using the invariants of the equation and controlling the extrema of the solution, it is demonstrated that the shapes of these elliptic periodic peakons are stable under small perturbations in the energy space. Throughout the paper, the theory of elliptic functions and elliptic integrals is used in the calculation.

Citation: 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
References:
[1]

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Second edition, Die Grundlehren der mathematischen Wissenschaften, Band 67 Springer-Verlag, New York-Heidelberg, 1971.

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

[4]

R. M. Chen, X. C. Liu, Y. Liu and C. Z. Qu, Stability of the Camassa-Holm peakons in the dynamics of a shallow-water-type system, Calc. Var. Partial Differential Equations, 55 (2016), Art. 34, 22 pp. doi: 10.1007/s00526-016-0972-0.

[5]

R. M. ChenJ. Lenells and Y. Liu, Stability of the $\mu$-Camassa-Holm Peakons, J. Nonlinear Sci., 23 (2013), 97-112.  doi: 10.1007/s00332-012-9141-6.

[6]

A. Chen, T. Deng and W. Huang, Orbital stability of trigonometric periodic peakons for the modified Camassa-Holm equation, Preprint, 2019.

[7]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.

[8]

A. Constantin, Global existence of solutions and wave breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble)., 50 (2000), 321-362. 

[9]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa., 26 (1998), 303-328. 

[10]

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>3.0.CO;2-L.

[11]

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.

[12]

A. Darós and L. K. Arruda, On the instability of elliptic traveling wave solutions of the modified Camassa-Holm equation, J. Differential Equeations, 266 (2019), 1946-1968.  doi: 10.1016/j.jde.2018.08.017.

[13]

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

[14]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Backlund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[15]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.

[16]

Z. H. GuoX. C. LiuX. X. Liu and C. Z. 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.

[17]

S. Hakkaev, I. D. Iliev and K. Kirchev, Stability of periodic travelling shallow-water waves determined by Newton's equation, J. Phys. A: Math. Theor., 41 (2008), 085203, 31 pp. doi: 10.1088/1751-8113/41/8/085203.

[18]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balance in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380.  doi: 10.1137/S1111111102410943.

[19]

B. KhesinJ. Lenells and G. Misiolek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342 (2008), 617-656.  doi: 10.1007/s00208-008-0250-3.

[20]

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

[21]

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

[22]

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

[23]

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.

[24]

X. C. LiuY. Liu and C. Z. 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.

[25]

X. C. LiuL. Yue and C. Z. Qu, Stability of peakons for the Novikov equation, J. Math. Pure Appl., 101 (2014), 17-187.  doi: 10.1016/j.matpur.2013.05.007.

[26]

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

[27]

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

[28]

C. Z. QuY. ZhangX. C. Liu and Y. Liu, Orbital stability of periodic peakons to a generalized $\mu$-Camassa-Holm equation, Arch. Rational Mech. Anal., 211 (2014), 593-617.  doi: 10.1007/s00205-013-0672-2.

[29]

Y. Wang and L. X. Tian, Stability of periodic peakons for the Novikov equation, (2018), arXiv: 1811.05835.

[30]

J. L. Yin, L. X. Tian and X. H. Fan, Stability of negative solitary waves for an integrable modified Camassa-Holm equation, J. Math. Phys., 51 (2010), 053515, 6 pp. doi: 10.1063/1.3407598.

show all references

References:
[1]

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Second edition, Die Grundlehren der mathematischen Wissenschaften, Band 67 Springer-Verlag, New York-Heidelberg, 1971.

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

[4]

R. M. Chen, X. C. Liu, Y. Liu and C. Z. Qu, Stability of the Camassa-Holm peakons in the dynamics of a shallow-water-type system, Calc. Var. Partial Differential Equations, 55 (2016), Art. 34, 22 pp. doi: 10.1007/s00526-016-0972-0.

[5]

R. M. ChenJ. Lenells and Y. Liu, Stability of the $\mu$-Camassa-Holm Peakons, J. Nonlinear Sci., 23 (2013), 97-112.  doi: 10.1007/s00332-012-9141-6.

[6]

A. Chen, T. Deng and W. Huang, Orbital stability of trigonometric periodic peakons for the modified Camassa-Holm equation, Preprint, 2019.

[7]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.

[8]

A. Constantin, Global existence of solutions and wave breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble)., 50 (2000), 321-362. 

[9]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa., 26 (1998), 303-328. 

[10]

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>3.0.CO;2-L.

[11]

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.

[12]

A. Darós and L. K. Arruda, On the instability of elliptic traveling wave solutions of the modified Camassa-Holm equation, J. Differential Equeations, 266 (2019), 1946-1968.  doi: 10.1016/j.jde.2018.08.017.

[13]

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

[14]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Backlund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[15]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.

[16]

Z. H. GuoX. C. LiuX. X. Liu and C. Z. 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.

[17]

S. Hakkaev, I. D. Iliev and K. Kirchev, Stability of periodic travelling shallow-water waves determined by Newton's equation, J. Phys. A: Math. Theor., 41 (2008), 085203, 31 pp. doi: 10.1088/1751-8113/41/8/085203.

[18]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balance in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380.  doi: 10.1137/S1111111102410943.

[19]

B. KhesinJ. Lenells and G. Misiolek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342 (2008), 617-656.  doi: 10.1007/s00208-008-0250-3.

[20]

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

[21]

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

[22]

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

[23]

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.

[24]

X. C. LiuY. Liu and C. Z. 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.

[25]

X. C. LiuL. Yue and C. Z. Qu, Stability of peakons for the Novikov equation, J. Math. Pure Appl., 101 (2014), 17-187.  doi: 10.1016/j.matpur.2013.05.007.

[26]

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

[27]

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

[28]

C. Z. QuY. ZhangX. C. Liu and Y. Liu, Orbital stability of periodic peakons to a generalized $\mu$-Camassa-Holm equation, Arch. Rational Mech. Anal., 211 (2014), 593-617.  doi: 10.1007/s00205-013-0672-2.

[29]

Y. Wang and L. X. Tian, Stability of periodic peakons for the Novikov equation, (2018), arXiv: 1811.05835.

[30]

J. L. Yin, L. X. Tian and X. H. Fan, Stability of negative solitary waves for an integrable modified Camassa-Holm equation, J. Math. Phys., 51 (2010), 053515, 6 pp. doi: 10.1063/1.3407598.

Figure 1.  (a) Phase portrait of the system (2.5) (b) The algebraic curve defined by H(φ, y) = 0
Figure 2.  The profile of elliptic periodic peakon for c = 1
[1]

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

[2]

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

[3]

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

[4]

Qiaoyi Hu, Zhijun Qiao. Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2613-2625. doi: 10.3934/dcds.2016.36.2613

[5]

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

[6]

Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019

[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]

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

[9]

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

[10]

Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047

[11]

Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883

[12]

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

[13]

Milena Stanislavova, Atanas Stefanov. Attractors for the viscous Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 159-186. doi: 10.3934/dcds.2007.18.159

[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]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (411)
  • HTML views (108)
  • Cited by (1)

Other articles
by authors

[Back to Top]