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

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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

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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

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