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Entire solutions with merging fronts to a bistable periodic lattice dynamical system
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 |
References:
[1] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[2] |
A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[3] |
R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[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), 167-178.
doi: 10.1023/B:JODY.0000041284.26400.d0. |
[5] |
K. S. Chou and C. Z. Qu, Integrable equations arising from motions of plane curves I, Physica D, 162 (2002), 9-33.
doi: 10.1016/S0167-2789(01)00364-5. |
[6] |
A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[7] |
A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[8] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[9] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[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), 303-328. |
[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), 475-504.
doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. |
[12] |
A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[13] |
A. Constantin and H. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.
doi: 10.1007/s00014-003-0785-6. |
[14] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[15] |
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. |
[16] |
A. Constantin and W. A. 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. |
[17] |
A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148.
doi: 10.1016/S0375-9601(00)00255-3. |
[18] |
H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207.
doi: 10.1007/BF01170373. |
[19] |
A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and perturbation theory (ed. A. Degasperis & G. Gaeta), pp 23-37, World Scientific, Singapore, 1999. |
[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), 87-117.
doi: 10.1512/iumj.2007.56.3040. |
[21] |
A. S. Fokas, On a class of physically important integrable equation, Physica D, 87 (1995), 145-150.
doi: 10.1016/0167-2789(95)00133-O. |
[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), 1905-1938.
doi: 10.1016/j.jde.2013.05.024. |
[23] |
B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 229-243.
doi: 10.1016/0167-2789(96)00048-6. |
[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. |
[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), 731-759.
doi: 10.1007/s00220-012-1566-0. |
[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), 505-523. |
[27] |
D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Analysis, 70 (2009), 1565-1573.
doi: 10.1016/j.na.2008.02.104. |
[28] |
S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868.
doi: 10.1063/1.532690. |
[29] |
J. Lenells, A variational approach to the stability of periodic peakons, J. Nonl. Math. Phys., 11 (2004), 151-163.
doi: 10.2991/jnmp.2004.11.2.2. |
[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), 27-63.
doi: 10.1006/jdeq.1999.3683. |
[31] |
Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.
doi: 10.1007/s00220-006-0082-5. |
[32] |
G. Misołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.
doi: 10.1016/S0393-0440(97)00010-7. |
[33] |
V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14pp.
doi: 10.1088/1751-8113/42/34/342002. |
[34] |
P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
[35] |
G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.
doi: 10.1016/S0362-546X(01)00791-X. |
[36] |
Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701,9pp.
doi: 10.1063/1.2365758. |
[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), 967-997.
doi: 10.1007/s00220-013-1749-3. |
[38] |
J. F. Toland, Stokes waves, Topol, Methods Nonlinear Anal., 7 (1996), 1-48. |
[39] |
Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation, Dyn. Cont. Discrete Impuls. Syst. Ser. A, Math. Anal., 12 (2005), 375-381. |
[40] |
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. |
[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), 1815-1844.
doi: 10.1081/PDE-120016129. |
show all references
References:
[1] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[2] |
A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[3] |
R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[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), 167-178.
doi: 10.1023/B:JODY.0000041284.26400.d0. |
[5] |
K. S. Chou and C. Z. Qu, Integrable equations arising from motions of plane curves I, Physica D, 162 (2002), 9-33.
doi: 10.1016/S0167-2789(01)00364-5. |
[6] |
A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[7] |
A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[8] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[9] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[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), 303-328. |
[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), 475-504.
doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. |
[12] |
A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[13] |
A. Constantin and H. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.
doi: 10.1007/s00014-003-0785-6. |
[14] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[15] |
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. |
[16] |
A. Constantin and W. A. 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. |
[17] |
A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148.
doi: 10.1016/S0375-9601(00)00255-3. |
[18] |
H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207.
doi: 10.1007/BF01170373. |
[19] |
A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and perturbation theory (ed. A. Degasperis & G. Gaeta), pp 23-37, World Scientific, Singapore, 1999. |
[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), 87-117.
doi: 10.1512/iumj.2007.56.3040. |
[21] |
A. S. Fokas, On a class of physically important integrable equation, Physica D, 87 (1995), 145-150.
doi: 10.1016/0167-2789(95)00133-O. |
[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), 1905-1938.
doi: 10.1016/j.jde.2013.05.024. |
[23] |
B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 229-243.
doi: 10.1016/0167-2789(96)00048-6. |
[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. |
[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), 731-759.
doi: 10.1007/s00220-012-1566-0. |
[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), 505-523. |
[27] |
D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Analysis, 70 (2009), 1565-1573.
doi: 10.1016/j.na.2008.02.104. |
[28] |
S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868.
doi: 10.1063/1.532690. |
[29] |
J. Lenells, A variational approach to the stability of periodic peakons, J. Nonl. Math. Phys., 11 (2004), 151-163.
doi: 10.2991/jnmp.2004.11.2.2. |
[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), 27-63.
doi: 10.1006/jdeq.1999.3683. |
[31] |
Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.
doi: 10.1007/s00220-006-0082-5. |
[32] |
G. Misołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.
doi: 10.1016/S0393-0440(97)00010-7. |
[33] |
V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14pp.
doi: 10.1088/1751-8113/42/34/342002. |
[34] |
P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
[35] |
G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.
doi: 10.1016/S0362-546X(01)00791-X. |
[36] |
Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701,9pp.
doi: 10.1063/1.2365758. |
[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), 967-997.
doi: 10.1007/s00220-013-1749-3. |
[38] |
J. F. Toland, Stokes waves, Topol, Methods Nonlinear Anal., 7 (1996), 1-48. |
[39] |
Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation, Dyn. Cont. Discrete Impuls. Syst. Ser. A, Math. Anal., 12 (2005), 375-381. |
[40] |
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. |
[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), 1815-1844.
doi: 10.1081/PDE-120016129. |
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