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, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013
References:
[1]

Ann. Inst. Fourier (Grenobite), 16 (1966), 319-361. doi: 10.5802/aif.233.  Google Scholar

[2]

Phys. Rev. Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[3]

Physica D, 162 (2002), 9-33. doi: 10.1016/S0167-2789(01)00364-5.  Google Scholar

[4]

J. Differential Equations, 246 (2009), 929-963. doi: 10.1016/j.jde.2008.04.014.  Google Scholar

[5]

J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.  Google Scholar

[6]

Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[7]

Bull. Amer. Math. Soc. (N. S.), 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[8]

Ann. of Math. (2), 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[9]

Ann. Global Anal. Geom., 31 (2007), 155-180. doi: 10.1007/s10455-006-9042-8.  Google Scholar

[10]

Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.  Google Scholar

[11]

J. Nonlin. Sci., 16 (2006), 109-122. doi: 10.1007/s00332-005-0707-4.  Google Scholar

[12]

Arch. Rational Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar

[13]

Comm. Pure Appl. Math., 52 (1999), 949-982.  Google Scholar

[14]

Physica D, 157 (2001), 75-89. doi: 10.1016/S0167-2789(01)00298-6.  Google Scholar

[15]

Physica D, 162 (2002), 9-33. doi: 10.1016/S0167-2789(01)00364-5.  Google Scholar

[16]

Comm. Pure Appl. Math., 53 (2000), 603-610.  Google Scholar

[17]

in Symmetry and Perturbation Theory (eds. A. Degasperis and G. Gaeta) (Rome, 1998), World Scientific Publ., River Edge, NJ, 1999, 23-37.  Google Scholar

[18]

Math. Z., 269 (2011), 1137-1153. doi: 10.1007/s00209-010-0778-2.  Google Scholar

[19]

J. Math. Phys., 51 (2010), 053101, 6 pp. 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]

Int. Math. Res. Notes, 2011 (2011), 2617-2649. doi: 10.1093/imrn/rnq163.  Google Scholar

[22]

SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075.  Google Scholar

[23]

Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.  Google Scholar

[24]

Int. Math. Res. Not., 2004 (2004), 485-499. doi: 10.1155/S1073792804132431.  Google Scholar

[25]

J. Nonlinear Math. Phys., 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2.  Google Scholar

[26]

Math. Ann., 342 (2008), 617-656. doi: 10.1007/s00208-008-0250-3.  Google Scholar

[27]

Adv. Math., 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.  Google Scholar

[28]

J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690.  Google Scholar

[29]

J. Geom. Phys., 57 (2007), 2049-2064. doi: 10.1016/j.geomphys.2007.05.003.  Google Scholar

[30]

2nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.  Google Scholar

[31]

J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7.  Google Scholar

[32]

Funktsional. Anal. i Prilozhen., 21 (1987), 81-82.  Google Scholar

[33]

Lett. Math. Phys., 59 (2002), 117-131. doi: 10.1023/A:1014933316169.  Google Scholar

[34]

J. Differential Equations, 254 (2013), 648-659. doi: 10.1016/j.jde.2012.09.012.  Google Scholar

show all references

References:
[1]

Ann. Inst. Fourier (Grenobite), 16 (1966), 319-361. doi: 10.5802/aif.233.  Google Scholar

[2]

Phys. Rev. Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[3]

Physica D, 162 (2002), 9-33. doi: 10.1016/S0167-2789(01)00364-5.  Google Scholar

[4]

J. Differential Equations, 246 (2009), 929-963. doi: 10.1016/j.jde.2008.04.014.  Google Scholar

[5]

J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.  Google Scholar

[6]

Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[7]

Bull. Amer. Math. Soc. (N. S.), 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[8]

Ann. of Math. (2), 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[9]

Ann. Global Anal. Geom., 31 (2007), 155-180. doi: 10.1007/s10455-006-9042-8.  Google Scholar

[10]

Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.  Google Scholar

[11]

J. Nonlin. Sci., 16 (2006), 109-122. doi: 10.1007/s00332-005-0707-4.  Google Scholar

[12]

Arch. Rational Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar

[13]

Comm. Pure Appl. Math., 52 (1999), 949-982.  Google Scholar

[14]

Physica D, 157 (2001), 75-89. doi: 10.1016/S0167-2789(01)00298-6.  Google Scholar

[15]

Physica D, 162 (2002), 9-33. doi: 10.1016/S0167-2789(01)00364-5.  Google Scholar

[16]

Comm. Pure Appl. Math., 53 (2000), 603-610.  Google Scholar

[17]

in Symmetry and Perturbation Theory (eds. A. Degasperis and G. Gaeta) (Rome, 1998), World Scientific Publ., River Edge, NJ, 1999, 23-37.  Google Scholar

[18]

Math. Z., 269 (2011), 1137-1153. doi: 10.1007/s00209-010-0778-2.  Google Scholar

[19]

J. Math. Phys., 51 (2010), 053101, 6 pp. 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]

Int. Math. Res. Notes, 2011 (2011), 2617-2649. doi: 10.1093/imrn/rnq163.  Google Scholar

[22]

SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075.  Google Scholar

[23]

Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.  Google Scholar

[24]

Int. Math. Res. Not., 2004 (2004), 485-499. doi: 10.1155/S1073792804132431.  Google Scholar

[25]

J. Nonlinear Math. Phys., 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2.  Google Scholar

[26]

Math. Ann., 342 (2008), 617-656. doi: 10.1007/s00208-008-0250-3.  Google Scholar

[27]

Adv. Math., 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.  Google Scholar

[28]

J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690.  Google Scholar

[29]

J. Geom. Phys., 57 (2007), 2049-2064. doi: 10.1016/j.geomphys.2007.05.003.  Google Scholar

[30]

2nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.  Google Scholar

[31]

J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7.  Google Scholar

[32]

Funktsional. Anal. i Prilozhen., 21 (1987), 81-82.  Google Scholar

[33]

Lett. Math. Phys., 59 (2002), 117-131. doi: 10.1023/A:1014933316169.  Google Scholar

[34]

J. Differential Equations, 254 (2013), 648-659. doi: 10.1016/j.jde.2012.09.012.  Google Scholar

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