American Institute of Mathematical Sciences

Advanced
August  2018, 38(8): 4163-4187. doi: 10.3934/dcds.2018181

On the Cauchy problem for a higher-order μ-Camassa-Holm equation

Feng Wang 1, , Fengquan Li 2, and  Zhijun Qiao 3,

1. 

School of Mathematics and Statistics, Xidian University, Xi'an 710071, China
2. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
3. 

School of Mathematical & Statistical Sciences, University of Texas-Rio Grande Valley, Texas 78539, USA

Authors to whom correspondence should be addressed. E-mails: wangfeng@xidian.edu.cn, fqli@dlut.edu.cn, zhijun.qiao@utrgv.edu.

Received  December 2017 Revised  February 2018 Published  May 2018

In this paper, we study the Cauchy problem of a higher-order μ-Camassa-Holm equation. We first establish the Green's function of $(μ-\partial_{x}^{2}+\partial_{x}^{4})^{-1}$ and local well-posedness for the equation in Sobolev spaces $H^{s}(\mathbb{S})$, $s>\frac{7}{2}$. Then we provide the global existence results for strong solutions and weak solutions. Moreover, we show that the solution map is non-uniformly continuous in $H^{s}(\mathbb{S})$, $s≥ 4$. Finally, we prove that the equation admits single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.

Keywords: Higher-order μ-Camassa-Holm equation, global existence, weak solutions, non-uniformly continuous, peakon solutions.
Mathematics Subject Classification: 35G25, 35L05, 35B30.
Citation: Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181
References:
[1]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857.

[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. Chen, J. Lenells and Y. Liu, Stability of the μ-Camassa-Holm peakons, J. Nonlinear Sci., 23 (2013), 97-112. doi: 10.1007/s00332-012-9141-6.

[4]

G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711.

[5]

G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations, J. Diff. Equ., 246 (2009), 929-963. doi: 10.1016/j.jde.2008.04.014.

[6]

G. M. Coclite and K. H. Karlsen, A note on the Camassa-Holm equation, J. Diff. Equ., 259 (2015), 2158-2166. doi: 10.1016/j.jde.2015.03.020.

[7]

G. M. Coclite and L. Ruvo, A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law, Discrete Contin. Dyn. Syst., 37 (2017), 1247-1282. doi: 10.3934/dcds.2017052.

[8]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Diff. Equ., 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333.

[9]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.

[10]

A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10 (2000), 391-399. doi: 10.1007/s003329910017.

[11]

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

[12]

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

[13]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.

[14]

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.

[15]

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.

[16]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[17]

A. Constantin and L. Molinet, Global weak solution solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801.

[18]

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.

[19]

R. Danchin, A few remarks on the Camassa-Holm equation, Diff. Integ. Equ., 14 (2001), 953-988.

[20]

D. Ding, Traveling solutions and evolution properties of the higher order Camassa-Holm equation, Nonlinear Anal., 152 (2017), 1-11. doi: 10.1016/j.na.2016.12.010.

[21]

D. Ding and P. Lv, Conservative solutions for higher-order Camassa-Holm equations, J. Math.Phys., 51 (2010), 072701, 15pp. doi: 10.1063/1.3462917.

[22]

D. Ding and S. Zhang, Lipschitz metric for the periodic second-order Camassa-Holm equation, J. Math. Anal. Appl., 451 (2017), 990-1025. doi: 10.1016/j.jmaa.2017.02.018.

[23]

J. Escher and B. Kolev, Geodesic completeness for Sobolev H^{s}$-metrics on the diffeomorphism group of the circle, J. Evol. Equ., 14 (2014), 949-968. doi: 10.1007/s00028-014-0245-3.

[24]

J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, J. Geom. Mech., 6 (2014), 335-372. doi: 10.3934/jgm.2014.6.335.

[25]

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

[26]

A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Diff. Int. Eqs., 22 (2009), 201-224.

[27]

A. Himonas, C. Kenig and G. Misiolek, Non-uniform dependence for the periodic CH equation, Comm. Partial Differential Equations, 35 (2010), 1145-1162. doi: 10.1080/03605300903436746.

[28]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, In: Spectral Theory and Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 448 (1975), 25-70.

[29]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 203-208. doi: 10.1002/cpa.3160410704.

[30]

B. Khesin, J. 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.

[31]

B. Kolev, Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst., 19 (2007), 555-574. doi: 10.3934/dcds.2007.19.555.

[32]

J. Lenells, G. Misiolek and F. Ti${\rm{\ddot g}}$ay, 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.

[33]

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

[34]

J. Liu and Z. Yin, On the Cauchy problem of a weakly dissipative μ-Hunter-Saxton equation, Ann. I. H. Poincaré-AN., 31 (2014), 267-279. doi: 10.1016/j.anihpc.2013.02.008.

[35]

R. McLachlan and X. Zhang, Well-posedness of a modified Camassa-Holm equations, J. Diff. Equ., 246 (2009), 3241-3259. doi: 10.1016/j.jde.2009.01.039.

[36]

Z. Qiao, The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Commun. Math. Phys., 239 (2003), 309-341. doi: 10.1007/s00220-003-0880-y.

[37]

J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[38] M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991. doi: 10.1007/978-1-4612-0431-2.
[39]

M. Taylor, Commutator estimates, Proc. Amer. Math. Soc., 131 (2003), 1501-1507. doi: 10.1090/S0002-9939-02-06723-0.

[40]

L. Tian, P. Zhang and L. Xia, Global existence for the higher-order Camassa-Holm shallow water equation, Nonlinear Anal., 74 (2011), 2468-2474. doi: 10.1016/j.na.2010.12.002.

[41]

F. Wang, F. Li and Z. Qiao, Well-posedness and peakons for a higher-order µ-Camassa-Holm equation, arXiv: 1712.07996.
[42]

S. Wu and Z. Yin, Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation, J. Diff. Equ., 246 (2009), 4309-4321. doi: 10.1016/j.jde.2008.12.008.

[43]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

show all references

References:
[1]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857.

[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. Chen, J. Lenells and Y. Liu, Stability of the μ-Camassa-Holm peakons, J. Nonlinear Sci., 23 (2013), 97-112. doi: 10.1007/s00332-012-9141-6.

[4]

G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711.

[5]

G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations, J. Diff. Equ., 246 (2009), 929-963. doi: 10.1016/j.jde.2008.04.014.

[6]

G. M. Coclite and K. H. Karlsen, A note on the Camassa-Holm equation, J. Diff. Equ., 259 (2015), 2158-2166. doi: 10.1016/j.jde.2015.03.020.

[7]

G. M. Coclite and L. Ruvo, A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law, Discrete Contin. Dyn. Syst., 37 (2017), 1247-1282. doi: 10.3934/dcds.2017052.

[8]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Diff. Equ., 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333.

[9]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.

[10]

A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10 (2000), 391-399. doi: 10.1007/s003329910017.

[11]

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

[12]

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

[13]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.

[14]

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.

[15]

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.

[16]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[17]

A. Constantin and L. Molinet, Global weak solution solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801.

[18]

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.

[19]

R. Danchin, A few remarks on the Camassa-Holm equation, Diff. Integ. Equ., 14 (2001), 953-988.

[20]

D. Ding, Traveling solutions and evolution properties of the higher order Camassa-Holm equation, Nonlinear Anal., 152 (2017), 1-11. doi: 10.1016/j.na.2016.12.010.

[21]

D. Ding and P. Lv, Conservative solutions for higher-order Camassa-Holm equations, J. Math.Phys., 51 (2010), 072701, 15pp. doi: 10.1063/1.3462917.

[22]

D. Ding and S. Zhang, Lipschitz metric for the periodic second-order Camassa-Holm equation, J. Math. Anal. Appl., 451 (2017), 990-1025. doi: 10.1016/j.jmaa.2017.02.018.

[23]

J. Escher and B. Kolev, Geodesic completeness for Sobolev H^{s}$-metrics on the diffeomorphism group of the circle, J. Evol. Equ., 14 (2014), 949-968. doi: 10.1007/s00028-014-0245-3.

[24]

J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, J. Geom. Mech., 6 (2014), 335-372. doi: 10.3934/jgm.2014.6.335.

[25]

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

[26]

A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Diff. Int. Eqs., 22 (2009), 201-224.

[27]

A. Himonas, C. Kenig and G. Misiolek, Non-uniform dependence for the periodic CH equation, Comm. Partial Differential Equations, 35 (2010), 1145-1162. doi: 10.1080/03605300903436746.

[28]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, In: Spectral Theory and Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 448 (1975), 25-70.

[29]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 203-208. doi: 10.1002/cpa.3160410704.

[30]

B. Khesin, J. 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.

[31]

B. Kolev, Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst., 19 (2007), 555-574. doi: 10.3934/dcds.2007.19.555.

[32]

J. Lenells, G. Misiolek and F. Ti${\rm{\ddot g}}$ay, 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.

[33]

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

[34]

J. Liu and Z. Yin, On the Cauchy problem of a weakly dissipative μ-Hunter-Saxton equation, Ann. I. H. Poincaré-AN., 31 (2014), 267-279. doi: 10.1016/j.anihpc.2013.02.008.

[35]

R. McLachlan and X. Zhang, Well-posedness of a modified Camassa-Holm equations, J. Diff. Equ., 246 (2009), 3241-3259. doi: 10.1016/j.jde.2009.01.039.

[36]

Z. Qiao, The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Commun. Math. Phys., 239 (2003), 309-341. doi: 10.1007/s00220-003-0880-y.

[37]

J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[38] M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991. doi: 10.1007/978-1-4612-0431-2.
[39]

M. Taylor, Commutator estimates, Proc. Amer. Math. Soc., 131 (2003), 1501-1507. doi: 10.1090/S0002-9939-02-06723-0.

[40]

L. Tian, P. Zhang and L. Xia, Global existence for the higher-order Camassa-Holm shallow water equation, Nonlinear Anal., 74 (2011), 2468-2474. doi: 10.1016/j.na.2010.12.002.

[41]

F. Wang, F. Li and Z. Qiao, Well-posedness and peakons for a higher-order µ-Camassa-Holm equation, arXiv: 1712.07996.
[42]

S. Wu and Z. Yin, Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation, J. Diff. Equ., 246 (2009), 4309-4321. doi: 10.1016/j.jde.2008.12.008.

[43]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

[1]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
[2]

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

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629
[4]

Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45
[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]

Qiaoyi Hu, Zhijun Qiao. Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2613-2625. doi: 10.3934/dcds.2016.36.2613
[7]

Xingxing Liu. Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5505-5521. doi: 10.3934/dcds.2018242
[8]

Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483
[9]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194
[10]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781
[11]

Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713
[12]

Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047
[13]

Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230
[14]

Alberto Bressan, Geng Chen, Qingtian Zhang. Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 25-42. doi: 10.3934/dcds.2015.35.25
[15]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064
[16]

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 & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019
[17]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[18]

Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059
[19]

Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459
[20]

Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (76)
  • HTML views (94)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences