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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

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
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