October  2015, 35(10): 5153-5169. doi: 10.3934/dcds.2015.35.5153

On the Cauchy problem for a four-component Camassa-Holm type system

1. 

Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

2. 

Department of Mathematics, Zhongshan University, Guangzhou, 510275

Received  December 2014 Revised  January 2015 Published  April 2015

This paper is concerned with a four-component Camassa-Holm type system proposed in [37], where its bi-Hamiltonian structure and infinitely many conserved quantities were constructed. In the paper, we first establish the local well-posedness for the system. Then we present several global existence and blow-up results for two integrable two-component subsystems.
Citation: Zeng Zhang, Zhaoyang Yin. On the Cauchy problem for a four-component Camassa-Holm type system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5153-5169. doi: 10.3934/dcds.2015.35.5153
References:
[1]

M. S. Alber, R. Camassa, D. D. Holm and J. E. Marsden, The geometry of peaked solitons and billiard solutions of a class of integrable PDEs,, Lett. Math. Phys., 32 (1994), 137.  doi: 10.1007/BF00739423.  Google Scholar

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[3]

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

[4]

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

[5]

C. 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

[6]

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.  Google Scholar

[7]

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

[8]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953.  doi: 10.1098/rspa.2000.0701.  Google Scholar

[9]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation,, Indiana Univ. Math. J., 47 (1998), 1527.   Google Scholar

[14]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423.  doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[15]

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

[16]

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

[17]

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.  Google Scholar

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.   Google Scholar

[24]

A. Degasperis, D. D. Kholm and A. N. I. Khon, A new integrable equation with peakon solutions,, Teoret. Mat. Fiz., 133 (2002), 170.  doi: 10.1023/A:1021186408422.  Google Scholar

[25]

A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and Perturbation Theory (Rome, (1998), 23.   Google Scholar

[26]

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

[27]

J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations,, J. Funct. Anal., 256 (2009), 479.  doi: 10.1016/j.jfa.2008.07.010.  Google Scholar

[28]

J. Escher and Z. Yin, Initial boundary value problems of the Degasperis-Procesi equation,, Polish Acad. Sci. Inst. Math., 81 (2008), 157.  doi: 10.4064/bc81-0-10.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

G. Gui, Y. Liu, P. J. Olver and C. 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

[33]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation,, Nonlinearity, 25 (2012), 449.  doi: 10.1088/0951-7715/25/2/449.  Google Scholar

[34]

D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: Geometrical aspects, peakons and numerical examples,, J. Phys. A, 43 (2010).  doi: 10.1088/1751-8113/43/49/492001.  Google Scholar

[35]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys. A, 41 (2008).  doi: 10.1088/1751-8113/41/37/372002.  Google Scholar

[36]

H. Li, Y. Li and Y. Chen, Bi-Hamiltonian structure of multi-component Novikov equation,, J. Nonlinear Math. Phys., 21 (2014), 509.  doi: 10.1080/14029251.2014.975522.  Google Scholar

[37]

N. Li, Q. P. Liu and Z. Popowicz, A four-component Camassa-Holm type hierarchy,, J. Geom. Phys., 85 (2014), 29.  doi: 10.1016/j.geomphys.2014.05.026.  Google Scholar

[38]

Y. A. Li and P. J. 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

[39]

X. Liu, Z. Qiao and Z. Yin, On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity,, Commun. Pure Appl. Anal., 13 (2014), 1283.  doi: 10.3934/cpaa.2014.13.1283.  Google Scholar

[40]

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

[41]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation,, J. Nonlinear Sci., 17 (2007), 169.  doi: 10.1007/s00332-006-0803-3.  Google Scholar

[42]

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

[43]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E (3), 53 (1996), 1900.  doi: 10.1103/PhysRevE.53.1900.  Google Scholar

[44]

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

[45]

Z. Qiao and B. Xia, Integrable peakon systems with weak kink and kink-peakon interactional solutions,, Front. Math. China, 8 (2013), 1185.  doi: 10.1007/s11464-013-0314-x.  Google Scholar

[46]

C. Qu, J. Song and R. Yao, Multi-component integrable systems and invariant curve flows in certain geometries,, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013).   Google Scholar

[47]

G. Rodríguez-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

[48]

J. Song, C. Qu and Z. Qiao, A new integrable two-component system with cubic nonlinearity,, J. Math. Phys., 52 (2011).  doi: 10.1063/1.3530865.  Google Scholar

[49]

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

[50]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation,, J. Phys. A, 44 (2011).  doi: 10.1088/1751-8113/44/5/055202.  Google Scholar

[51]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 707.   Google Scholar

[52]

B. Xia and Z. Qiao, A new two-component integrable system with peakon and weak kink solutions,, preprint, ().   Google Scholar

[53]

B. Xia and Z. Qiao, Integrable multi-component Camassa-Holm system,, preprint, ().   Google Scholar

[54]

B. Xia, Z. Qiao and R. Zhou, A synthetical integrable two-component model with peakon solutions,, preprint, ().   Google Scholar

[55]

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

[56]

K. Yan, Z. Qiao and Z. Yin, Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions,, Comm. Math. Phys., 336 (2015), 581.  doi: 10.1007/s00220-014-2236-1.  Google Scholar

[57]

Z. Yin, Global weak solutions for a new periodic integrable equation with peakon solutions,, J. Funct. Anal., 212 (2004), 182.  doi: 10.1016/j.jfa.2003.07.010.  Google Scholar

[58]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions,, Illinois J. Math., 47 (2003), 649.   Google Scholar

[59]

Z. Zhang and Z. Yin, Well-posedness, global existence and blow-up phenomena for an integrable multi-component Camassa-Holm system,, preprint, ().   Google Scholar

show all references

References:
[1]

M. S. Alber, R. Camassa, D. D. Holm and J. E. Marsden, The geometry of peaked solitons and billiard solutions of a class of integrable PDEs,, Lett. Math. Phys., 32 (1994), 137.  doi: 10.1007/BF00739423.  Google Scholar

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[3]

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

[4]

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

[5]

C. 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

[6]

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.  Google Scholar

[7]

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

[8]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953.  doi: 10.1098/rspa.2000.0701.  Google Scholar

[9]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation,, Indiana Univ. Math. J., 47 (1998), 1527.   Google Scholar

[14]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423.  doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[15]

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

[16]

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

[17]

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.  Google Scholar

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.   Google Scholar

[24]

A. Degasperis, D. D. Kholm and A. N. I. Khon, A new integrable equation with peakon solutions,, Teoret. Mat. Fiz., 133 (2002), 170.  doi: 10.1023/A:1021186408422.  Google Scholar

[25]

A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and Perturbation Theory (Rome, (1998), 23.   Google Scholar

[26]

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

[27]

J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations,, J. Funct. Anal., 256 (2009), 479.  doi: 10.1016/j.jfa.2008.07.010.  Google Scholar

[28]

J. Escher and Z. Yin, Initial boundary value problems of the Degasperis-Procesi equation,, Polish Acad. Sci. Inst. Math., 81 (2008), 157.  doi: 10.4064/bc81-0-10.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

G. Gui, Y. Liu, P. J. Olver and C. 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

[33]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation,, Nonlinearity, 25 (2012), 449.  doi: 10.1088/0951-7715/25/2/449.  Google Scholar

[34]

D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: Geometrical aspects, peakons and numerical examples,, J. Phys. A, 43 (2010).  doi: 10.1088/1751-8113/43/49/492001.  Google Scholar

[35]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys. A, 41 (2008).  doi: 10.1088/1751-8113/41/37/372002.  Google Scholar

[36]

H. Li, Y. Li and Y. Chen, Bi-Hamiltonian structure of multi-component Novikov equation,, J. Nonlinear Math. Phys., 21 (2014), 509.  doi: 10.1080/14029251.2014.975522.  Google Scholar

[37]

N. Li, Q. P. Liu and Z. Popowicz, A four-component Camassa-Holm type hierarchy,, J. Geom. Phys., 85 (2014), 29.  doi: 10.1016/j.geomphys.2014.05.026.  Google Scholar

[38]

Y. A. Li and P. J. 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

[39]

X. Liu, Z. Qiao and Z. Yin, On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity,, Commun. Pure Appl. Anal., 13 (2014), 1283.  doi: 10.3934/cpaa.2014.13.1283.  Google Scholar

[40]

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

[41]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation,, J. Nonlinear Sci., 17 (2007), 169.  doi: 10.1007/s00332-006-0803-3.  Google Scholar

[42]

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

[43]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E (3), 53 (1996), 1900.  doi: 10.1103/PhysRevE.53.1900.  Google Scholar

[44]

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

[45]

Z. Qiao and B. Xia, Integrable peakon systems with weak kink and kink-peakon interactional solutions,, Front. Math. China, 8 (2013), 1185.  doi: 10.1007/s11464-013-0314-x.  Google Scholar

[46]

C. Qu, J. Song and R. Yao, Multi-component integrable systems and invariant curve flows in certain geometries,, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013).   Google Scholar

[47]

G. Rodríguez-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

[48]

J. Song, C. Qu and Z. Qiao, A new integrable two-component system with cubic nonlinearity,, J. Math. Phys., 52 (2011).  doi: 10.1063/1.3530865.  Google Scholar

[49]

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

[50]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation,, J. Phys. A, 44 (2011).  doi: 10.1088/1751-8113/44/5/055202.  Google Scholar

[51]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 707.   Google Scholar

[52]

B. Xia and Z. Qiao, A new two-component integrable system with peakon and weak kink solutions,, preprint, ().   Google Scholar

[53]

B. Xia and Z. Qiao, Integrable multi-component Camassa-Holm system,, preprint, ().   Google Scholar

[54]

B. Xia, Z. Qiao and R. Zhou, A synthetical integrable two-component model with peakon solutions,, preprint, ().   Google Scholar

[55]

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

[56]

K. Yan, Z. Qiao and Z. Yin, Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions,, Comm. Math. Phys., 336 (2015), 581.  doi: 10.1007/s00220-014-2236-1.  Google Scholar

[57]

Z. Yin, Global weak solutions for a new periodic integrable equation with peakon solutions,, J. Funct. Anal., 212 (2004), 182.  doi: 10.1016/j.jfa.2003.07.010.  Google Scholar

[58]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions,, Illinois J. Math., 47 (2003), 649.   Google Scholar

[59]

Z. Zhang and Z. Yin, Well-posedness, global existence and blow-up phenomena for an integrable multi-component Camassa-Holm system,, preprint, ().   Google Scholar

[1]

Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171

[2]

Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112

[3]

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

[4]

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

[5]

Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025

[6]

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

[7]

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

[8]

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

[9]

Kai Yan, Zhaoyang Yin. Well-posedness for a modified two-component Camassa-Holm system in critical spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1699-1712. doi: 10.3934/dcds.2013.33.1699

[10]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139

[11]

Zeng Zhang, Zhaoyang Yin. Global existence for a two-component Camassa-Holm system with an arbitrary smooth function. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5523-5536. doi: 10.3934/dcds.2018243

[12]

Katrin Grunert. Blow-up for the two-component Camassa--Holm system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2041-2051. doi: 10.3934/dcds.2015.35.2041

[13]

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

[14]

Chenghua Wang, Rong Zeng, Shouming Zhou, Bin Wang, Chunlai Mu. Continuity for the rotation-two-component Camassa-Holm system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6633-6652. doi: 10.3934/dcdsb.2019160

[15]

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

[16]

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

[17]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809

[18]

Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203

[19]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111

[20]

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

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]