December  2016, 36(12): 7235-7256. doi: 10.3934/dcds.2016115

Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion

1. 

Department of Mathematics, Nanjing Forestry University, Nanjing 210036

2. 

Department of Mathematics, Southwest University, Chongqing 400715

Received  December 2015 Revised  January 2016 Published  October 2016

Considered herein is the blow-up mechanism to the periodic modified Camassa-Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result. The key feature is the ratio between solution and its gradient. Using the continuity of the solutions and the right transformation, we then obtain this blow-up criterion to the case with negative linear dispersion and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that when the linear dispersion is non-negative, formation of singularity can be induced by an initial datum with a sufficiently steep profile.
Citation: 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
References:
[1]

L. Brandolese, Local-in-space criteria for blow-up in shallow water and dispersive rod equations,, Comm. Math. Phys., 330 (2014), 401. doi: 10.1007/s00220-014-1958-4. Google Scholar

[2]

L. Brandolese and M. F. Cortez, Blowup issues for a class of nonlinear dispersive wave equations,, J. Differential Equations, 256 (2014), 3981. doi: 10.1016/j.jde.2014.03.008. Google Scholar

[3]

L. Brandolese and M. F. Cortez, On permanent and breading waves in hyperelastic rods and rings,, J. Funct. Anal., 266 (2014), 6954. doi: 10.1016/j.jfa.2014.02.039. Google Scholar

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar

[5]

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

[6]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar

[7]

M. Chen, Y. Liu, C. Qu and S. Zhang, Oscillatio-induced blow-up to the modified Camassa-Holm equation with linear dispersion,, Adv. Math., 272 (2015), 225. doi: 10.1016/j.aim.2014.12.003. Google Scholar

[8]

K. S. Chou and C. Z. Qu, Integrable equations arising from motions of plane curves I,, Physica D, 162 (2002), 9. doi: 10.1016/S0167-2789(01)00364-5. Google Scholar

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar

[10]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. Roy. Soc. London A, 457 (2001), 953. doi: 10.1098/rspa.2000.0701. Google Scholar

[11]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar

[12]

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

[13]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 26 (1998), 303. Google Scholar

[14]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12. Google Scholar

[16]

A. Constantin and H. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comment. Math. Helv., 78 (2003), 787. doi: 10.1007/s00014-003-0785-6. Google Scholar

[17]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar

[18]

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

[19]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar

[20]

A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods,, Phys. Lett. A, 270 (2000), 140. doi: 10.1016/S0375-9601(00)00255-3. Google Scholar

[21]

H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod,, Acta Mech., 127 (1998), 193. doi: 10.1007/BF01170373. Google Scholar

[22]

A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and perturbation theory (ed. A. Degasperis & G. Gaeta), (1999), 23. Google Scholar

[23]

J. Escher, Y. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation,, Indiana Univ. Math. J., 56 (2007), 87. doi: 10.1512/iumj.2007.56.3040. Google Scholar

[24]

A. S. Fokas, On a class of physically important integrable equation,, Physica D, 87 (1995), 145. doi: 10.1016/0167-2789(95)00133-O. Google Scholar

[25]

Y. Fu, G. L. Gui, Y. Liu and C. Z. Qu, On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity,, J. Differential Equations, 255 (2013), 1905. doi: 10.1016/j.jde.2013.05.024. Google Scholar

[26]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation,, Physica D, 95 (1996), 229. doi: 10.1016/0167-2789(96)00048-6. Google Scholar

[27]

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

[28]

G. L. Gui, Y. Liu , P. Olver and C. Z. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation,, Comm. Math. Phys., 319 (2013), 731. doi: 10.1007/s00220-012-1566-0. Google Scholar

[29]

H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons,, Discrete Contin. Dyn. Syst., 14 (2006), 505. doi: 10.3934/dcds.2006.14.505. Google Scholar

[30]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690. Google Scholar

[31]

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

[32]

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

[33]

Y. Liu, P. Olver, C. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation,, Anal. Appl. (Singap.), 12 (2014), 355. doi: 10.1142/S0219530514500274. Google Scholar

[34]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801. doi: 10.1007/s00220-006-0082-5. Google Scholar

[35]

Y.Matsuno, Smooth and singular multisolution solutions of a modified Camassa-Holm equation with cubic nonlinearity and linear dispersion,, J. Phys. A., 47 (2014). doi: 10.1088/1751-8113/47/12/125203. Google Scholar

[36]

G. Misołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar

[37]

V. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/34/342002. Google Scholar

[38]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900. Google Scholar

[39]

Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons,, J. Math. Phys., 47 (2006). doi: 10.1063/1.2365758. Google Scholar

[40]

Z. Qiao and X. Q. Li, An integrable equation with nonsmooth solitons,, Theor. Math. Phys., 167 (2011), 584. doi: 10.1007/s11232-011-0044-8. Google Scholar

[41]

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

[42]

G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar

[43]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1. Google Scholar

[44]

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

[45]

Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation,, Comm. Partial Differential Equations, 27 (2002), 1815. doi: 10.1081/PDE-120016129. Google Scholar

[46]

Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation,, Dyn. Cont. Discrete Impuls. Syst. Ser. A, 12 (2005), 375. Google Scholar

[47]

M. Zhu and S. Zhang, On the blow-up of solutions to the periodic modified integrable Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 36 (2016), 2347. doi: 10.3934/dcds.2016.36.2347. Google Scholar

show all references

References:
[1]

L. Brandolese, Local-in-space criteria for blow-up in shallow water and dispersive rod equations,, Comm. Math. Phys., 330 (2014), 401. doi: 10.1007/s00220-014-1958-4. Google Scholar

[2]

L. Brandolese and M. F. Cortez, Blowup issues for a class of nonlinear dispersive wave equations,, J. Differential Equations, 256 (2014), 3981. doi: 10.1016/j.jde.2014.03.008. Google Scholar

[3]

L. Brandolese and M. F. Cortez, On permanent and breading waves in hyperelastic rods and rings,, J. Funct. Anal., 266 (2014), 6954. doi: 10.1016/j.jfa.2014.02.039. Google Scholar

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar

[5]

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

[6]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar

[7]

M. Chen, Y. Liu, C. Qu and S. Zhang, Oscillatio-induced blow-up to the modified Camassa-Holm equation with linear dispersion,, Adv. Math., 272 (2015), 225. doi: 10.1016/j.aim.2014.12.003. Google Scholar

[8]

K. S. Chou and C. Z. Qu, Integrable equations arising from motions of plane curves I,, Physica D, 162 (2002), 9. doi: 10.1016/S0167-2789(01)00364-5. Google Scholar

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar

[10]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. Roy. Soc. London A, 457 (2001), 953. doi: 10.1098/rspa.2000.0701. Google Scholar

[11]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar

[12]

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

[13]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 26 (1998), 303. Google Scholar

[14]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12. Google Scholar

[16]

A. Constantin and H. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comment. Math. Helv., 78 (2003), 787. doi: 10.1007/s00014-003-0785-6. Google Scholar

[17]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar

[18]

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

[19]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar

[20]

A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods,, Phys. Lett. A, 270 (2000), 140. doi: 10.1016/S0375-9601(00)00255-3. Google Scholar

[21]

H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod,, Acta Mech., 127 (1998), 193. doi: 10.1007/BF01170373. Google Scholar

[22]

A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and perturbation theory (ed. A. Degasperis & G. Gaeta), (1999), 23. Google Scholar

[23]

J. Escher, Y. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation,, Indiana Univ. Math. J., 56 (2007), 87. doi: 10.1512/iumj.2007.56.3040. Google Scholar

[24]

A. S. Fokas, On a class of physically important integrable equation,, Physica D, 87 (1995), 145. doi: 10.1016/0167-2789(95)00133-O. Google Scholar

[25]

Y. Fu, G. L. Gui, Y. Liu and C. Z. Qu, On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity,, J. Differential Equations, 255 (2013), 1905. doi: 10.1016/j.jde.2013.05.024. Google Scholar

[26]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation,, Physica D, 95 (1996), 229. doi: 10.1016/0167-2789(96)00048-6. Google Scholar

[27]

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

[28]

G. L. Gui, Y. Liu , P. Olver and C. Z. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation,, Comm. Math. Phys., 319 (2013), 731. doi: 10.1007/s00220-012-1566-0. Google Scholar

[29]

H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons,, Discrete Contin. Dyn. Syst., 14 (2006), 505. doi: 10.3934/dcds.2006.14.505. Google Scholar

[30]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690. Google Scholar

[31]

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

[32]

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

[33]

Y. Liu, P. Olver, C. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation,, Anal. Appl. (Singap.), 12 (2014), 355. doi: 10.1142/S0219530514500274. Google Scholar

[34]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801. doi: 10.1007/s00220-006-0082-5. Google Scholar

[35]

Y.Matsuno, Smooth and singular multisolution solutions of a modified Camassa-Holm equation with cubic nonlinearity and linear dispersion,, J. Phys. A., 47 (2014). doi: 10.1088/1751-8113/47/12/125203. Google Scholar

[36]

G. Misołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar

[37]

V. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/34/342002. Google Scholar

[38]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900. Google Scholar

[39]

Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons,, J. Math. Phys., 47 (2006). doi: 10.1063/1.2365758. Google Scholar

[40]

Z. Qiao and X. Q. Li, An integrable equation with nonsmooth solitons,, Theor. Math. Phys., 167 (2011), 584. doi: 10.1007/s11232-011-0044-8. Google Scholar

[41]

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

[42]

G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar

[43]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1. Google Scholar

[44]

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

[45]

Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation,, Comm. Partial Differential Equations, 27 (2002), 1815. doi: 10.1081/PDE-120016129. Google Scholar

[46]

Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation,, Dyn. Cont. Discrete Impuls. Syst. Ser. A, 12 (2005), 375. Google Scholar

[47]

M. Zhu and S. Zhang, On the blow-up of solutions to the periodic modified integrable Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 36 (2016), 2347. doi: 10.3934/dcds.2016.36.2347. Google Scholar

[1]

Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027

[2]

Min Zhu, Shuanghu Zhang. On the blow-up of solutions to the periodic modified integrable Camassa--Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2347-2364. doi: 10.3934/dcds.2016.36.2347

[3]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501

[4]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067

[5]

Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493

[6]

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

[7]

Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042

[8]

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

[9]

Ying Fu. A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2011-2039. doi: 10.3934/dcds.2015.35.2011

[10]

Guangying Lv, Mingxin Wang. Some remarks for a modified periodic Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1161-1180. doi: 10.3934/dcds.2011.30.1161

[11]

Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065

[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]

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

[14]

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

[15]

Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871

[16]

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

[17]

Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499

[18]

Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389

[19]

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

[20]

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

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]