July  2014, 34(7): 3013-3024. doi: 10.3934/dcds.2014.34.3013

On the higher-order b-family equation and Euler equations on the circle

1. 

Department of Mathematics, Nanjing Forestry University, Nanjing 210037, China

Received  June 2013 Revised  September 2013 Published  December 2013

Considered herein is a geometric investigation on the higher-order b-family equation describing exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane. It is shown that the higher-order $b-$family equation can only be realized as an Euler equation on the Lie group Diff$(\mathbb{S}^1) $ of all smooth and orientation preserving diffeomorphisms on the circle if the parameter $b=2$ which corresponds to the higher-order Camassa-Holm equation with the metric $H^k, k\ge 1. $
Citation: Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013
References:
[1]

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenobite), 16 (1966), 319.  doi: 10.5802/aif.233.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations,, J. Differential Equations, 246 (2009), 929.  doi: 10.1016/j.jde.2008.04.014.  Google Scholar

[5]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352.  doi: 10.1006/jfan.1997.3231.  Google Scholar

[6]

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

[7]

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

[8]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math. (2), 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[9]

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

[10]

A. Constantin and B. 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

[11]

A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle,, J. Nonlin. Sci., 16 (2006), 109.  doi: 10.1007/s00332-005-0707-4.  Google Scholar

[12]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rational Mech. Anal., 192 (2009), 165.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[13]

A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1999), 949.   Google Scholar

[14]

A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation,, Physica D, 157 (2001), 75.  doi: 10.1016/S0167-2789(01)00298-6.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation,, Math. Z., 269 (2011), 1137.  doi: 10.1007/s00209-010-0778-2.  Google Scholar

[19]

J. Escher and J. Seiler, The periodic $b$-equations and Euler equations on the circle,, J. Math. Phys., 51 (2010).  doi: 10.1063/1.3405494.  Google Scholar

[20]

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

[21]

P. Górka and E. G. Reyes, The modified Camassa-Holm equation,, Int. Math. Res. Notes, 2011 (2011), 2617.  doi: 10.1093/imrn/rnq163.  Google Scholar

[22]

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

[23]

R. Ivanov, Water waves and integrability,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2267.  doi: 10.1098/rsta.2007.2007.  Google Scholar

[24]

J. Lenells, Stability of periodic peakons,, Int. Math. Res. Not., 2004 (2004), 485.  doi: 10.1155/S1073792804132431.  Google Scholar

[25]

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

[26]

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

[27]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116.  doi: 10.1016/S0001-8708(02)00063-4.  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, The Hunter-Saxton equation describes the geodesic flow on a sphere,, J. Geom. Phys., 57 (2007), 2049.  doi: 10.1016/j.geomphys.2007.05.003.  Google Scholar

[30]

J. E. Marsden and T. Ratiu, Introcudction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,, 2nd edition, (1999).   Google Scholar

[31]

G. Misiołek, 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

[32]

V. Yu. Ovsienko and B. Khesin, The super Korteweg-de Vries equation as an Euler equation,, Funktsional. Anal. i Prilozhen., 21 (1987), 81.   Google Scholar

[33]

E. G. Reyes, Geometric integrability of the Camassa-Holm equation,, Lett. Math. Phys., 59 (2002), 117.  doi: 10.1023/A:1014933316169.  Google Scholar

[34]

M. Zhu, Y. Liu and C. Z. Qu, On the model of the compressible hyperelastic rods and Euler equations on the circle,, J. Differential Equations, 254 (2013), 648.  doi: 10.1016/j.jde.2012.09.012.  Google Scholar

show all references

References:
[1]

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenobite), 16 (1966), 319.  doi: 10.5802/aif.233.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations,, J. Differential Equations, 246 (2009), 929.  doi: 10.1016/j.jde.2008.04.014.  Google Scholar

[5]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352.  doi: 10.1006/jfan.1997.3231.  Google Scholar

[6]

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

[7]

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

[8]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math. (2), 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[9]

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

[10]

A. Constantin and B. 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

[11]

A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle,, J. Nonlin. Sci., 16 (2006), 109.  doi: 10.1007/s00332-005-0707-4.  Google Scholar

[12]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rational Mech. Anal., 192 (2009), 165.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[13]

A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1999), 949.   Google Scholar

[14]

A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation,, Physica D, 157 (2001), 75.  doi: 10.1016/S0167-2789(01)00298-6.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation,, Math. Z., 269 (2011), 1137.  doi: 10.1007/s00209-010-0778-2.  Google Scholar

[19]

J. Escher and J. Seiler, The periodic $b$-equations and Euler equations on the circle,, J. Math. Phys., 51 (2010).  doi: 10.1063/1.3405494.  Google Scholar

[20]

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

[21]

P. Górka and E. G. Reyes, The modified Camassa-Holm equation,, Int. Math. Res. Notes, 2011 (2011), 2617.  doi: 10.1093/imrn/rnq163.  Google Scholar

[22]

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

[23]

R. Ivanov, Water waves and integrability,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2267.  doi: 10.1098/rsta.2007.2007.  Google Scholar

[24]

J. Lenells, Stability of periodic peakons,, Int. Math. Res. Not., 2004 (2004), 485.  doi: 10.1155/S1073792804132431.  Google Scholar

[25]

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

[26]

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

[27]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116.  doi: 10.1016/S0001-8708(02)00063-4.  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, The Hunter-Saxton equation describes the geodesic flow on a sphere,, J. Geom. Phys., 57 (2007), 2049.  doi: 10.1016/j.geomphys.2007.05.003.  Google Scholar

[30]

J. E. Marsden and T. Ratiu, Introcudction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,, 2nd edition, (1999).   Google Scholar

[31]

G. Misiołek, 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

[32]

V. Yu. Ovsienko and B. Khesin, The super Korteweg-de Vries equation as an Euler equation,, Funktsional. Anal. i Prilozhen., 21 (1987), 81.   Google Scholar

[33]

E. G. Reyes, Geometric integrability of the Camassa-Holm equation,, Lett. Math. Phys., 59 (2002), 117.  doi: 10.1023/A:1014933316169.  Google Scholar

[34]

M. Zhu, Y. Liu and C. Z. Qu, On the model of the compressible hyperelastic rods and Euler equations on the circle,, J. Differential Equations, 254 (2013), 648.  doi: 10.1016/j.jde.2012.09.012.  Google Scholar

[1]

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

[2]

Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047

[3]

Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883

[4]

Milena Stanislavova, Atanas Stefanov. Attractors for the viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 159-186. doi: 10.3934/dcds.2007.18.159

[5]

Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871

[6]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067

[7]

Yongsheng Mi, Boling Guo, Chunlai Mu. Persistence properties for the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019243

[8]

Stephen C. Anco, Elena Recio, María L. Gandarias, María S. Bruzón. A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Conference Publications, 2015, 2015 (special) : 29-37. doi: 10.3934/proc.2015.0029

[9]

Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230

[10]

Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305

[11]

Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713

[12]

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

[13]

Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181

[14]

Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483

[15]

Priscila Leal da Silva, Igor Leite Freire. An equation unifying both Camassa-Holm and Novikov equations. Conference Publications, 2015, 2015 (special) : 304-311. doi: 10.3934/proc.2015.0304

[16]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194

[17]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629

[18]

Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45

[19]

Alberto Bressan, Geng Chen, Qingtian Zhang. Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 25-42. doi: 10.3934/dcds.2015.35.25

[20]

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

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]