April  2016, 36(4): 2347-2364. doi: 10.3934/dcds.2016.36.2347

On the blow-up of solutions to the periodic modified integrable Camassa--Holm equation

1. 

Department of Mathematics, Nanjing Forestry University, Nanjing 210036, China

2. 

Department of Mathematics, Southwest University, Chongqing 400715, China

Received  March 2015 Revised  April 2015 Published  September 2015

We derive conditions on the initial data, including cases where the initial momentum density is not of one sign, that produce blow-up of the induced solution to the periodic modified Camassa-Holm equation with cubic nonlinearity. The blow-up conditions and the blow-up rate are formulated in terms of the initial momentum density and the average initial energy.
Citation: 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
References:
[1]

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

[2]

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

[3]

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

[4]

C. S. Cao, D. D. Holm and E. S. Titi, Traveling wave solutions for a class of one-dimensional nonlinear shallow water wave models,, J. Dynam. Differential Equations, 16 (2004), 167. doi: 10.1023/B:JODY.0000041284.26400.d0. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

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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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons,, Disc. Cont. Dyn. Syst. A., 14 (2006), 505. Google Scholar

[27]

D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlinear Analysis, 70 (2009), 1565. doi: 10.1016/j.na.2008.02.104. Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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 (2000), 1815. doi: 10.1081/PDE-120016129. Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

C. S. Cao, D. D. Holm and E. S. Titi, Traveling wave solutions for a class of one-dimensional nonlinear shallow water wave models,, J. Dynam. Differential Equations, 16 (2004), 167. doi: 10.1023/B:JODY.0000041284.26400.d0. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons,, Disc. Cont. Dyn. Syst. A., 14 (2006), 505. Google Scholar

[27]

D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlinear Analysis, 70 (2009), 1565. doi: 10.1016/j.na.2008.02.104. Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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 (2000), 1815. doi: 10.1081/PDE-120016129. Google Scholar

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