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The Aubry-Mather theorem for driven generalized elastic chains
On the higher-order b-family equation and Euler equations on the circle
1. | Department of Mathematics, Nanjing Forestry University, Nanjing 210037, China |
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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