October  2020, 40(10): 5929-5954. doi: 10.3934/dcds.2020253

The two-component $ \mu $-Camassa–Holm system with peaked solutions

1. 

School of Mathematics and Statistics and Center for Nonlinear Studies, Ningbo University, Ningbo 315211, China

2. 

School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127, China

Received  December 2019 Revised  May 2020 Published  June 2020

Fund Project: The work of Li is partially supported by the NSF-China grant-11971251. The work of Fu is partially supported by the NSF-China grant-11631007 and grant-11471259, and the National Science Basic Research Program of Shaanxi (Program No. 2019JM-007 and 2020JC-37). The work of Qu is partially supported by the NSF-China grant-11631007 and grant-11971251

This paper is mainly concerned with the classification of the general two-component $ \mu $-Camassa-Holm systems with quadratic nonlinearities. As a conclusion of such classification, a two-component $ \mu $-Camassa-Holm system admitting multi-peaked solutions and $ H^1 $-norm conservation law is found, which is a $ \mu $-version of the two-component modified Camassa-Holm system and can be derived from the semidirect-product Euler-Poincaré equations corresponding to a Lagrangian. The local well-posedness for solutions to the initial value problem associated with the two-component $ \mu $-Camassa-Holm system is established. And the precise blow-up scenario, wave breaking phenomena and blow-up rate for solutions of this problem are also investigated.

Citation: Yingying Li, Ying Fu, Changzheng Qu. The two-component $ \mu $-Camassa–Holm system with peaked solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5929-5954. doi: 10.3934/dcds.2020253
References:
[1]

M. S. AlberR. CamassaD. 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), 137-151.  doi: 10.1007/BF00739423.  Google Scholar

[2]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[3]

C. CaoD. 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.  Google Scholar

[4]

R. M. ChenJ. Lenells and Y. Liu, Stability of the $\mu$-Camassa–Holm peakons, J. Nonlinear Sci., 23 (2013), 97-112.  doi: 10.1007/s00332-012-9141-6.  Google Scholar

[5]

M. ChenS. Liu and Y. Zhang, A two-component generalization of the Camassa–Holm equation and its solutions, Lett. Math. Phys, 75 (2006), 1-15.  doi: 10.1007/s11005-005-0041-7.  Google Scholar

[6]

K.-S. Chou and C. Qu, Integrable equations arising from motions of plane curves, Phys. D, 162 (2002), 9-33.  doi: 10.1016/S0167-2789(01)00364-5.  Google Scholar

[7]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

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A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10 (2000), 391-399.  doi: 10.1007/s003329910017.  Google Scholar

[9]

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.  Google Scholar

[10]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[11]

A. ConstantinV. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.  doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[12]

A. ConstantinV. S. Gerdjikov and R. I. Ivanov, Generalised Fourier transform for the Camassa–Holm hierarchy, Inverse Problems, 23 (2007), 1565-1597.  doi: 10.1088/0266-5611/23/4/012.  Google Scholar

[13]

A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.  doi: 10.1016/j.physleta.2008.10.050.  Google Scholar

[14]

A. ConstantinT. KappelerB. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180.  doi: 10.1007/s10455-006-9042-8.  Google Scholar

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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.  Google Scholar

[16]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.   Google Scholar

[17]

K. El Dika and L. Molinet, Stability of multipeakons, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1517-1532.  doi: 10.1016/j.anihpc.2009.02.002.  Google Scholar

[18]

Y. FuY. Liu and C. Qu, Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons, Math. Ann., 348 (2010), 415-448.  doi: 10.1007/s00208-010-0483-9.  Google Scholar

[19]

Y. Fu and C. Qu, Well posedness and blow-up solution for a new coupled Camassa–Holm equations with peakons, J. Math. Phys., 50 (2009), 012906, 25 pp. doi: 10.1063/1.3064810.  Google Scholar

[20]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa–Holm equation, Phys. D, 95 (1996), 229-243.  doi: 10.1016/0167-2789(96)00048-6.  Google Scholar

[21]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[22]

K. GrunertH. Holden and X. Raynaud, Global solutions for the two-component Camassa–Holm system, Comm. Partial Differential Equations, 37 (2012), 2245-2271.  doi: 10.1080/03605302.2012.683505.  Google Scholar

[23]

C. Guan and Z. Yin, Global weak solutions for a two-component Camassa–Holm shallow water system, J. Funct. Anal., 260 (2011), 1132-1154.  doi: 10.1016/j.jfa.2010.11.015.  Google Scholar

[24]

C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa–Holm equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 623-641.  doi: 10.1016/j.anihpc.2011.04.003.  Google Scholar

[25]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa–Holm system, J. Funct. Anal., 258 (2010), 4251-4278.  doi: 10.1016/j.jfa.2010.02.008.  Google Scholar

[26]

D. D. Holm, L. Ó Náraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E, 79 (2009), 016601, 13 pp. doi: 10.1103/PhysRevE.79.016601.  Google Scholar

[27]

J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.  doi: 10.1137/0151075.  Google Scholar

[28] R. J. Iorio Jr. and V. de Magalhães Iorio, Fourier Analysis and Partial Differential Equations, Cambridge Studies in Advanced Mathematics, 70, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511623745.  Google Scholar
[29]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224.  Google Scholar

[30]

T. Kato and G. Ponce, Communtator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[31]

B. KhesinJ. Lenells and G. Misiolek, Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342 (2008), 617-656.  doi: 10.1007/s00208-008-0250-3.  Google Scholar

[32]

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.  Google Scholar

[33]

J. Lenells, The scattering approach for the Camassa-Holm equation, J. Nonlinear Math. Phys., 9 (2002), 389-393.  doi: 10.2991/jnmp.2002.9.4.2.  Google Scholar

[34]

J. LenellsG. Misiolek and F. Tiğlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys., 299 (2010), 129-161.  doi: 10.1007/s00220-010-1069-9.  Google Scholar

[35]

G. Lv and M. Wang, Some remarks for a modified periodic Camassa–Holm system, Discrete Contin. Dyn. Syst., 30 (2011), 1161-1180.  doi: 10.3934/dcds.2011.30.1161.  Google Scholar

[36]

G. Misiolek, 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.  Google Scholar

[37]

P. J. Olver, Invariant submanifold flows, J. Phys. A, 41 (2008), 344017, 22 pp. doi: 10.1088/1751-8113/41/34/344017.  Google Scholar

[38]

P. J. 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.  Google Scholar

[39]

J. Schiff, The Camassa–Holm equation: A loop group approach, Phys. D, 121 (1998), 24-43.  doi: 10.1016/S0167-2789(98)00099-2.  Google Scholar

[40]

W. Tan and Z. Yin, Global conservative solutions of a modified two-component Camassa–Holm shallow water system, J. Differential Equations, 251 (2011), 3558-3582.  doi: 10.1016/j.jde.2011.08.010.  Google Scholar

[41]

W. Tan and Z. Yin, Global periodic conservative solutions of a periodic modified two-component Camassa–Holm equation, J. Funct. Anal., 261 (2011), 1204-1226.  doi: 10.1016/j.jfa.2011.04.015.  Google Scholar

[42]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 8 (1996), 413-414.  doi: 10.12775/TMNA.1996.001.  Google Scholar

[43]

X. Wu and B. Guo, Persistence properties and infinite propagation for the modified periodic 2-component Camassa–Holm quation, Discrete Contin. Dyn. Syst., 33 (2013), 3211-3223.  doi: 10.3934/dcds.2013.33.3211.  Google Scholar

[44]

W. Yan and Y. Li, The Cauchy problem for the modified two-component Camassa–Holm system in critical Besov space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 443-469.  doi: 10.1016/j.anihpc.2014.01.003.  Google Scholar

[45]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.  doi: 10.1215/ijm/1258138186.  Google Scholar

[46]

D. Zuo, A two-component $\mu$-Hunter-Saxton equation, Inverse Problems, 26 (2010), 085003, 9 pp. doi: 10.1088/0266-5611/26/8/085003.  Google Scholar

show all references

References:
[1]

M. S. AlberR. CamassaD. 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), 137-151.  doi: 10.1007/BF00739423.  Google Scholar

[2]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[3]

C. CaoD. 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.  Google Scholar

[4]

R. M. ChenJ. Lenells and Y. Liu, Stability of the $\mu$-Camassa–Holm peakons, J. Nonlinear Sci., 23 (2013), 97-112.  doi: 10.1007/s00332-012-9141-6.  Google Scholar

[5]

M. ChenS. Liu and Y. Zhang, A two-component generalization of the Camassa–Holm equation and its solutions, Lett. Math. Phys, 75 (2006), 1-15.  doi: 10.1007/s11005-005-0041-7.  Google Scholar

[6]

K.-S. Chou and C. Qu, Integrable equations arising from motions of plane curves, Phys. D, 162 (2002), 9-33.  doi: 10.1016/S0167-2789(01)00364-5.  Google Scholar

[7]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[8]

A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10 (2000), 391-399.  doi: 10.1007/s003329910017.  Google Scholar

[9]

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.  Google Scholar

[10]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[11]

A. ConstantinV. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.  doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[12]

A. ConstantinV. S. Gerdjikov and R. I. Ivanov, Generalised Fourier transform for the Camassa–Holm hierarchy, Inverse Problems, 23 (2007), 1565-1597.  doi: 10.1088/0266-5611/23/4/012.  Google Scholar

[13]

A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.  doi: 10.1016/j.physleta.2008.10.050.  Google Scholar

[14]

A. ConstantinT. KappelerB. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180.  doi: 10.1007/s10455-006-9042-8.  Google Scholar

[15]

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.  Google Scholar

[16]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.   Google Scholar

[17]

K. El Dika and L. Molinet, Stability of multipeakons, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1517-1532.  doi: 10.1016/j.anihpc.2009.02.002.  Google Scholar

[18]

Y. FuY. Liu and C. Qu, Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons, Math. Ann., 348 (2010), 415-448.  doi: 10.1007/s00208-010-0483-9.  Google Scholar

[19]

Y. Fu and C. Qu, Well posedness and blow-up solution for a new coupled Camassa–Holm equations with peakons, J. Math. Phys., 50 (2009), 012906, 25 pp. doi: 10.1063/1.3064810.  Google Scholar

[20]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa–Holm equation, Phys. D, 95 (1996), 229-243.  doi: 10.1016/0167-2789(96)00048-6.  Google Scholar

[21]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[22]

K. GrunertH. Holden and X. Raynaud, Global solutions for the two-component Camassa–Holm system, Comm. Partial Differential Equations, 37 (2012), 2245-2271.  doi: 10.1080/03605302.2012.683505.  Google Scholar

[23]

C. Guan and Z. Yin, Global weak solutions for a two-component Camassa–Holm shallow water system, J. Funct. Anal., 260 (2011), 1132-1154.  doi: 10.1016/j.jfa.2010.11.015.  Google Scholar

[24]

C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa–Holm equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 623-641.  doi: 10.1016/j.anihpc.2011.04.003.  Google Scholar

[25]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa–Holm system, J. Funct. Anal., 258 (2010), 4251-4278.  doi: 10.1016/j.jfa.2010.02.008.  Google Scholar

[26]

D. D. Holm, L. Ó Náraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E, 79 (2009), 016601, 13 pp. doi: 10.1103/PhysRevE.79.016601.  Google Scholar

[27]

J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.  doi: 10.1137/0151075.  Google Scholar

[28] R. J. Iorio Jr. and V. de Magalhães Iorio, Fourier Analysis and Partial Differential Equations, Cambridge Studies in Advanced Mathematics, 70, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511623745.  Google Scholar
[29]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224.  Google Scholar

[30]

T. Kato and G. Ponce, Communtator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[31]

B. KhesinJ. Lenells and G. Misiolek, Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342 (2008), 617-656.  doi: 10.1007/s00208-008-0250-3.  Google Scholar

[32]

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.  Google Scholar

[33]

J. Lenells, The scattering approach for the Camassa-Holm equation, J. Nonlinear Math. Phys., 9 (2002), 389-393.  doi: 10.2991/jnmp.2002.9.4.2.  Google Scholar

[34]

J. LenellsG. Misiolek and F. Tiğlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys., 299 (2010), 129-161.  doi: 10.1007/s00220-010-1069-9.  Google Scholar

[35]

G. Lv and M. Wang, Some remarks for a modified periodic Camassa–Holm system, Discrete Contin. Dyn. Syst., 30 (2011), 1161-1180.  doi: 10.3934/dcds.2011.30.1161.  Google Scholar

[36]

G. Misiolek, 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.  Google Scholar

[37]

P. J. Olver, Invariant submanifold flows, J. Phys. A, 41 (2008), 344017, 22 pp. doi: 10.1088/1751-8113/41/34/344017.  Google Scholar

[38]

P. J. 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.  Google Scholar

[39]

J. Schiff, The Camassa–Holm equation: A loop group approach, Phys. D, 121 (1998), 24-43.  doi: 10.1016/S0167-2789(98)00099-2.  Google Scholar

[40]

W. Tan and Z. Yin, Global conservative solutions of a modified two-component Camassa–Holm shallow water system, J. Differential Equations, 251 (2011), 3558-3582.  doi: 10.1016/j.jde.2011.08.010.  Google Scholar

[41]

W. Tan and Z. Yin, Global periodic conservative solutions of a periodic modified two-component Camassa–Holm equation, J. Funct. Anal., 261 (2011), 1204-1226.  doi: 10.1016/j.jfa.2011.04.015.  Google Scholar

[42]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 8 (1996), 413-414.  doi: 10.12775/TMNA.1996.001.  Google Scholar

[43]

X. Wu and B. Guo, Persistence properties and infinite propagation for the modified periodic 2-component Camassa–Holm quation, Discrete Contin. Dyn. Syst., 33 (2013), 3211-3223.  doi: 10.3934/dcds.2013.33.3211.  Google Scholar

[44]

W. Yan and Y. Li, The Cauchy problem for the modified two-component Camassa–Holm system in critical Besov space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 443-469.  doi: 10.1016/j.anihpc.2014.01.003.  Google Scholar

[45]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.  doi: 10.1215/ijm/1258138186.  Google Scholar

[46]

D. Zuo, A two-component $\mu$-Hunter-Saxton equation, Inverse Problems, 26 (2010), 085003, 9 pp. doi: 10.1088/0266-5611/26/8/085003.  Google Scholar

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