American Institute of Mathematical Sciences

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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

Min Zhu 1,

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. $
Keywords: Euler equation, Camassa-Holm equation, b-family equation, diffeomorphisms group of the circle..
Mathematics Subject Classification: Primary: 35Q35, 58D0.
Citation: Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete and Continuous Dynamical Systems, 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), 319-361. doi: 10.5802/aif.233.

[2]

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

[3]

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

[4]

G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations, J. Differential Equations, 246 (2009), 929-963. doi: 10.1016/j.jde.2008.04.014.

[5]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.

[6]

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

[7]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. (N. S.), 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.

[8]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math. (2), 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.

[9]

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

[10]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.

[11]

A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle, J. Nonlin. Sci., 16 (2006), 109-122. doi: 10.1007/s00332-005-0707-4.

[12]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rational Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.

[13]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.

[14]

A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation, Physica D, 157 (2001), 75-89. doi: 10.1016/S0167-2789(01)00298-6.

[15]

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

[16]

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

[17]

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

[18]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153. doi: 10.1007/s00209-010-0778-2.

[19]

J. Escher and J. Seiler, The periodic $b$-equations and Euler equations on the circle, J. Math. Phys., 51 (2010), 053101, 6 pp. doi: 10.1063/1.3405494.

[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.

[21]

P. Górka and E. G. Reyes, The modified Camassa-Holm equation, Int. Math. Res. Notes, 2011 (2011), 2617-2649. doi: 10.1093/imrn/rnq163.

[22]

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

[23]

R. Ivanov, Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.

[24]

J. Lenells, Stability of periodic peakons, Int. Math. Res. Not., 2004 (2004), 485-499. doi: 10.1155/S1073792804132431.

[25]

J. Lenells, A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2.

[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), 617-656. doi: 10.1007/s00208-008-0250-3.

[27]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.

[28]

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.

[29]

J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049-2064. doi: 10.1016/j.geomphys.2007.05.003.

[30]

J. E. Marsden and T. Ratiu, Introcudction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.

[31]

G. Misiołek, 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.

[32]

V. Yu. Ovsienko and B. Khesin, The super Korteweg-de Vries equation as an Euler equation, Funktsional. Anal. i Prilozhen., 21 (1987), 81-82.

[33]

E. G. Reyes, Geometric integrability of the Camassa-Holm equation, Lett. Math. Phys., 59 (2002), 117-131. doi: 10.1023/A:1014933316169.

[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), 648-659. doi: 10.1016/j.jde.2012.09.012.

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), 319-361. doi: 10.5802/aif.233.

[2]

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

[3]

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

[4]

G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations, J. Differential Equations, 246 (2009), 929-963. doi: 10.1016/j.jde.2008.04.014.

[5]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.

[6]

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

[7]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. (N. S.), 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.

[8]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math. (2), 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.

[9]

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

[10]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.

[11]

A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle, J. Nonlin. Sci., 16 (2006), 109-122. doi: 10.1007/s00332-005-0707-4.

[12]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rational Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.

[13]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.

[14]

A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation, Physica D, 157 (2001), 75-89. doi: 10.1016/S0167-2789(01)00298-6.

[15]

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

[16]

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

[17]

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

[18]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153. doi: 10.1007/s00209-010-0778-2.

[19]

J. Escher and J. Seiler, The periodic $b$-equations and Euler equations on the circle, J. Math. Phys., 51 (2010), 053101, 6 pp. doi: 10.1063/1.3405494.

[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.

[21]

P. Górka and E. G. Reyes, The modified Camassa-Holm equation, Int. Math. Res. Notes, 2011 (2011), 2617-2649. doi: 10.1093/imrn/rnq163.

[22]

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

[23]

R. Ivanov, Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.

[24]

J. Lenells, Stability of periodic peakons, Int. Math. Res. Not., 2004 (2004), 485-499. doi: 10.1155/S1073792804132431.

[25]

J. Lenells, A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2.

[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), 617-656. doi: 10.1007/s00208-008-0250-3.

[27]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.

[28]

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.

[29]

J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049-2064. doi: 10.1016/j.geomphys.2007.05.003.

[30]

J. E. Marsden and T. Ratiu, Introcudction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.

[31]

G. Misiołek, 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.

[32]

V. Yu. Ovsienko and B. Khesin, The super Korteweg-de Vries equation as an Euler equation, Funktsional. Anal. i Prilozhen., 21 (1987), 81-82.

[33]

E. G. Reyes, Geometric integrability of the Camassa-Holm equation, Lett. Math. Phys., 59 (2002), 117-131. doi: 10.1023/A:1014933316169.

[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), 648-659. doi: 10.1016/j.jde.2012.09.012.

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