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Liouville type theorem for nonlinear elliptic equation with general nonlinearity
The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces
1. | College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China |
References:
[1] |
A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces,, SIAM Rev., 43 (2001), 585.
doi: 10.1137/S0036144501386986. |
[2] |
R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy,, Adv. Math., 140 (1998), 190.
doi: 10.1006/aima.1998.1768. |
[3] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Rat. Mech. Anal., 183 (2007), 215.
doi: 10.1007/s00205-006-0010-z. |
[4] |
L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces,, Int. Math. Res. Not., 22 (2012), 5161.
|
[5] |
A. Boutet de Monvel and D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line,, C. R. Math. Acad. Sci. Paris, 343 (2006), 627.
doi: 10.1016/j.crma.2006.10.014. |
[6] |
R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Letters, 71 (1993), 1661.
doi: 10.1103/PhysRevLett.71.1661. |
[7] |
R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1.
doi: 10.1016/S0065-2156(08)70254-0. |
[8] |
G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation,, J. Funct. Anal., 233 (2006), 60.
doi: 10.1016/j.jfa.2005.07.008. |
[9] |
A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352.
doi: 10.1006/jfan.1997.3231. |
[10] |
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. |
[11] |
A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. Roy. Soc. London, 457 (2001), 953.
doi: 10.1098/rspa.2000.0701. |
[12] |
A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 26 (1998), 303.
|
[13] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Mathematica, 181 (1998), 229.
doi: 10.1007/BF02392586. |
[14] |
A. Constantin and J. Escher, Global weak solutions for a shallow water equation,, Indiana. Univ. Math. J., 47 (1998), 1527.
doi: 10.1512/iumj.1998.47.1466. |
[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. |
[16] |
A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197.
doi: 10.1088/0266-5611/22/6/017. |
[17] |
A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129.
doi: 10.1016/j.physleta.2008.10.050. |
[18] |
A. Constantin, R. I. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation,, Nonlinearity, 23 (2010), 2559.
doi: 10.1088/0951-7715/23/10/012. |
[19] |
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. |
[20] |
A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45.
doi: 10.1007/s002200050801. |
[21] |
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. |
[22] |
A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, J. Nonlinear. Sci., 12 (2002), 415.
doi: 10.1007/s00332-002-0517-x. |
[23] |
R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.
|
[24] |
R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429.
doi: 10.1016/S0022-0396(03)00096-2. |
[25] |
R. Danchin, Fourier Analysis Methods for PDEs,, Lecture Notes, (2003). Google Scholar |
[26] |
A. Degasperis and M. Procesi, Asymptotic integrability, in: Symmetry and Perturbation Theory,, World Scientific, (1999), 23.
|
[27] |
A. Degasperis, D. Holm and A. Hone, A new integral equation with peakon solutions,, Theoret. Math. Phys., 133 (2002), 1463.
doi: 10.1023/A:1021186408422. |
[28] |
A. Degasperis, D. D. Holm and A. N. W. Hone, Integral and non-integrable equations with peakons,, Nonlinear physics: Theory and experiment, (2003), 37.
doi: 10.1142/9789812704467_0005. |
[29] |
H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid Dyn. Res., 33 (2003), 73.
doi: 10.1016/S0169-5983(03)00046-7. |
[30] |
H. R. Dullin, G. A. Gottwald and D. D. Holm, On asymptotically equivalent shallow water wave equations,, Phys. D., 190 (2004), 1.
doi: 10.1016/j.physd.2003.11.004. |
[31] |
H. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion,, Phys. Rev. Letters, 87 (2001), 4501.
doi: 10.1103/PhysRevLett.87.194501. |
[32] |
J. Escher, Y. Liu and Z. Y. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation,, J. Funct. Anal., 241 (2006), 457.
doi: 10.1016/j.jfa.2006.03.022. |
[33] |
J. Escher, Y. Liu, 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. |
[34] |
J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the $b$-equation,, J. reine Angew. Math., 624 (2008), 51.
doi: 10.1515/CRELLE.2008.080. |
[35] |
A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries,, Phys. D, 4 (): 47.
doi: 10.1016/0167-2789(81)90004-X. |
[36] |
K. Gröchenig, Weight functions in time-frequency analysis, Pseudo-differential operators: Partial differential equations and time-frequency analysis,, Fields Inst. Commun., 52 (2007), 343.
|
[37] |
G. L. Gui, Y. Liu and T. X. Tian, Global existence and blow-up phenomena for the peakon $b$-family of equations,, Indiana Univ. Math. J., 57 (2008), 1209.
doi: 10.1512/iumj.2008.57.3213. |
[38] |
D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlinear Anal., 70 (2009), 1565.
doi: 10.1016/j.na.2008.02.104. |
[39] |
A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation,, Nonlinearity, 25 (2012), 449.
doi: 10.1088/0951-7715/25/2/449. |
[40] |
A. A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation,, Comm. Math. Phy., 271 (2007), 511.
doi: 10.1007/s00220-006-0172-4. |
[41] |
D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs,, SIAM J. Appl. Dyn. Syst., 2 (2003), 323.
doi: 10.1137/S1111111102410943. |
[42] |
D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1-1 nonlinear evolutionary PDE,, Phys. Lett. A, 308 (2003), 437.
doi: 10.1016/S0375-9601(03)00114-2. |
[43] |
A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys. Appl. Math. Theor., 41 (2008).
doi: 10.1088/1751-8113/41/37/372002. |
[44] |
A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm equation,, Dyn. Partial Differ. Equ., 6 (2009), 253.
doi: 10.4310/DPDE.2009.v6.n3.a3. |
[45] |
R. I. Ivanov, Water waves and integrability,, Philos. Trans. Roy. Soc. London A, 365 (2007), 2267.
doi: 10.1098/rsta.2007.2007. |
[46] |
R. Ivanov, Extended Camassa-Holm hierarchy and conserved quantities,, Z. Naturforsch. A, 61 (2006), 133. Google Scholar |
[47] |
Z. H. Jiang and L. D. Ni, Blow-up phemomena for the integrable Novikov equation,, J. Math. Appl. Anal., 385 (2012), 551.
doi: 10.1016/j.jmaa.2011.06.067. |
[48] |
K. Grayshan, Peakon solutions of the Novikov equation and properties of the data-to-solution map,, J. Math. Anal. Appl., 397 (2013), 515.
doi: 10.1016/j.jmaa.2012.08.006. |
[49] |
S. Y Lai, N. Li and Y. H. Wu, The existence of global strong and weak solutions for the Novikov equation,, J. Math. Anal. Appl., 399 (2013), 682.
doi: 10.1016/j.jmaa.2012.10.048. |
[50] |
J. Lenells, Conservation laws of the Camassa-Holm equation,, J. Phys. (A), 38 (2005), 869.
doi: 10.1088/0305-4470/38/4/007. |
[51] |
Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Diff. Equ., 162 (2000), 27.
doi: 10.1006/jdeq.1999.3683. |
[52] |
N. Li, S. Y. Lai, S. Li and M. Wu, The local and global existence of solutions for a generalized Camasa-Holm equation,, Abstr. Appl. Anal., (2012).
doi: 10.1155/2012/532369. |
[53] |
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. |
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H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation,, Inverse Problems, 19 (2003), 1241.
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A. V. Mikhailov and V. S. Novikov, Perturbative symmetry approach,, J. Phys. (A), 35 (2002), 4775.
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L. D. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation,, J. Differential Equations, 250 (2011), 3002.
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L. D. Ni and Y. Zhou, A new asymptotic behavior of solutions to the Camassa-holm equation,, Proc. Amer. Math. Soc., 140 (2012), 607.
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W. Niu and S. Zhang, Blow-up phenomena and global existence for the nouniform weakly dissipative $b$-equation,, J. Math. Anal. Appl., 374 (2011), 166.
doi: 10.1016/j.jmaa.2010.08.002. |
[59] |
V. S. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A, 42 (2009).
doi: 10.1088/1751-8113/42/34/342002. |
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F. Tiglay, The periodic cauchy problem for novikov's equation,, Int. Math. Res. Not., 20 (2011), 4633.
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V. O. Vakhnenko and E. J. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation,, Chaos Solitons Fractals, 20 (2004), 1059.
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X. L. Wu and Z. Y. Yin, A note on the Cauchy problem of the Novikov equation,, Appl. Anal., 92 (2013), 1116.
doi: 10.1080/00036811.2011.649735. |
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S. Y. Wu and Z. Y. Yin, Global weak solutions for the Novikov equation,, J. Phys. A: Math. Theor., 44 (2011).
doi: 10.1088/1751-8113/44/5/055202. |
[64] |
Z. P. 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. |
[65] |
W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the integrable Novikov equation,, J. Differential Equations, 253 (2012), 298.
doi: 10.1016/j.jde.2012.03.015. |
[66] |
Z. Y. Yin, Global solutions to a new integrable equation with peakons,, Indiana. Univ. Math. J., 53 (2004), 1189.
doi: 10.1512/iumj.2004.53.2479. |
[67] |
Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions,, IIIinois J. Math., 47 (2003), 649.
|
[68] |
W. Yan, Y. S. Li and Y. M. Zhang, The cauchy problem for the Novikov equation,, Nonlinear Differ. Equ. Appl., 20 (2013), 1157.
doi: 10.1007/s00030-012-0202-1. |
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S. M. Zhou and C. L. Mu, The properties of solutions for a generalized $b$-family equation with higher-order nonlinearities and peakons,, J. Nonlinear Sci., 23 (2013), 863.
doi: 10.1007/s00332-013-9171-8. |
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S. M. Zhou, C. L. Mu and L. C. Wang, Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation,, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 843.
doi: 10.3934/dcds.2014.34.843. |
show all references
References:
[1] |
A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces,, SIAM Rev., 43 (2001), 585.
doi: 10.1137/S0036144501386986. |
[2] |
R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy,, Adv. Math., 140 (1998), 190.
doi: 10.1006/aima.1998.1768. |
[3] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Rat. Mech. Anal., 183 (2007), 215.
doi: 10.1007/s00205-006-0010-z. |
[4] |
L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces,, Int. Math. Res. Not., 22 (2012), 5161.
|
[5] |
A. Boutet de Monvel and D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line,, C. R. Math. Acad. Sci. Paris, 343 (2006), 627.
doi: 10.1016/j.crma.2006.10.014. |
[6] |
R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Letters, 71 (1993), 1661.
doi: 10.1103/PhysRevLett.71.1661. |
[7] |
R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1.
doi: 10.1016/S0065-2156(08)70254-0. |
[8] |
G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation,, J. Funct. Anal., 233 (2006), 60.
doi: 10.1016/j.jfa.2005.07.008. |
[9] |
A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352.
doi: 10.1006/jfan.1997.3231. |
[10] |
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. |
[11] |
A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. Roy. Soc. London, 457 (2001), 953.
doi: 10.1098/rspa.2000.0701. |
[12] |
A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 26 (1998), 303.
|
[13] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Mathematica, 181 (1998), 229.
doi: 10.1007/BF02392586. |
[14] |
A. Constantin and J. Escher, Global weak solutions for a shallow water equation,, Indiana. Univ. Math. J., 47 (1998), 1527.
doi: 10.1512/iumj.1998.47.1466. |
[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. |
[16] |
A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197.
doi: 10.1088/0266-5611/22/6/017. |
[17] |
A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129.
doi: 10.1016/j.physleta.2008.10.050. |
[18] |
A. Constantin, R. I. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation,, Nonlinearity, 23 (2010), 2559.
doi: 10.1088/0951-7715/23/10/012. |
[19] |
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. |
[20] |
A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45.
doi: 10.1007/s002200050801. |
[21] |
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. |
[22] |
A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, J. Nonlinear. Sci., 12 (2002), 415.
doi: 10.1007/s00332-002-0517-x. |
[23] |
R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.
|
[24] |
R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429.
doi: 10.1016/S0022-0396(03)00096-2. |
[25] |
R. Danchin, Fourier Analysis Methods for PDEs,, Lecture Notes, (2003). Google Scholar |
[26] |
A. Degasperis and M. Procesi, Asymptotic integrability, in: Symmetry and Perturbation Theory,, World Scientific, (1999), 23.
|
[27] |
A. Degasperis, D. Holm and A. Hone, A new integral equation with peakon solutions,, Theoret. Math. Phys., 133 (2002), 1463.
doi: 10.1023/A:1021186408422. |
[28] |
A. Degasperis, D. D. Holm and A. N. W. Hone, Integral and non-integrable equations with peakons,, Nonlinear physics: Theory and experiment, (2003), 37.
doi: 10.1142/9789812704467_0005. |
[29] |
H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid Dyn. Res., 33 (2003), 73.
doi: 10.1016/S0169-5983(03)00046-7. |
[30] |
H. R. Dullin, G. A. Gottwald and D. D. Holm, On asymptotically equivalent shallow water wave equations,, Phys. D., 190 (2004), 1.
doi: 10.1016/j.physd.2003.11.004. |
[31] |
H. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion,, Phys. Rev. Letters, 87 (2001), 4501.
doi: 10.1103/PhysRevLett.87.194501. |
[32] |
J. Escher, Y. Liu and Z. Y. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation,, J. Funct. Anal., 241 (2006), 457.
doi: 10.1016/j.jfa.2006.03.022. |
[33] |
J. Escher, Y. Liu, 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. |
[34] |
J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the $b$-equation,, J. reine Angew. Math., 624 (2008), 51.
doi: 10.1515/CRELLE.2008.080. |
[35] |
A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries,, Phys. D, 4 (): 47.
doi: 10.1016/0167-2789(81)90004-X. |
[36] |
K. Gröchenig, Weight functions in time-frequency analysis, Pseudo-differential operators: Partial differential equations and time-frequency analysis,, Fields Inst. Commun., 52 (2007), 343.
|
[37] |
G. L. Gui, Y. Liu and T. X. Tian, Global existence and blow-up phenomena for the peakon $b$-family of equations,, Indiana Univ. Math. J., 57 (2008), 1209.
doi: 10.1512/iumj.2008.57.3213. |
[38] |
D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlinear Anal., 70 (2009), 1565.
doi: 10.1016/j.na.2008.02.104. |
[39] |
A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation,, Nonlinearity, 25 (2012), 449.
doi: 10.1088/0951-7715/25/2/449. |
[40] |
A. A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation,, Comm. Math. Phy., 271 (2007), 511.
doi: 10.1007/s00220-006-0172-4. |
[41] |
D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs,, SIAM J. Appl. Dyn. Syst., 2 (2003), 323.
doi: 10.1137/S1111111102410943. |
[42] |
D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1-1 nonlinear evolutionary PDE,, Phys. Lett. A, 308 (2003), 437.
doi: 10.1016/S0375-9601(03)00114-2. |
[43] |
A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys. Appl. Math. Theor., 41 (2008).
doi: 10.1088/1751-8113/41/37/372002. |
[44] |
A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm equation,, Dyn. Partial Differ. Equ., 6 (2009), 253.
doi: 10.4310/DPDE.2009.v6.n3.a3. |
[45] |
R. I. Ivanov, Water waves and integrability,, Philos. Trans. Roy. Soc. London A, 365 (2007), 2267.
doi: 10.1098/rsta.2007.2007. |
[46] |
R. Ivanov, Extended Camassa-Holm hierarchy and conserved quantities,, Z. Naturforsch. A, 61 (2006), 133. Google Scholar |
[47] |
Z. H. Jiang and L. D. Ni, Blow-up phemomena for the integrable Novikov equation,, J. Math. Appl. Anal., 385 (2012), 551.
doi: 10.1016/j.jmaa.2011.06.067. |
[48] |
K. Grayshan, Peakon solutions of the Novikov equation and properties of the data-to-solution map,, J. Math. Anal. Appl., 397 (2013), 515.
doi: 10.1016/j.jmaa.2012.08.006. |
[49] |
S. Y Lai, N. Li and Y. H. Wu, The existence of global strong and weak solutions for the Novikov equation,, J. Math. Anal. Appl., 399 (2013), 682.
doi: 10.1016/j.jmaa.2012.10.048. |
[50] |
J. Lenells, Conservation laws of the Camassa-Holm equation,, J. Phys. (A), 38 (2005), 869.
doi: 10.1088/0305-4470/38/4/007. |
[51] |
Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Diff. Equ., 162 (2000), 27.
doi: 10.1006/jdeq.1999.3683. |
[52] |
N. Li, S. Y. Lai, S. Li and M. Wu, The local and global existence of solutions for a generalized Camasa-Holm equation,, Abstr. Appl. Anal., (2012).
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