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Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach
1. | Institute for Applied Mathematics, University of Hanover, D-30167 Hanover |
2. | School of Mathematical Sciences, University College Cork, Cork, Ireland |
References:
[1] |
V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319.
doi: 10.5802/aif.233. |
[2] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.
doi: 10.1103/PhysRevLett.71.1661. |
[3] |
R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1.
doi: 10.1016/S0065-2156(08)70254-0. |
[4] |
A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321.
doi: 10.5802/aif.1757. |
[5] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, Volume 81 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), (2011).
doi: 10.1137/1.9781611971873. |
[6] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.
doi: 10.1007/BF02392586. |
[7] |
A. Constantin and J. Escher, Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475.
doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. |
[8] |
A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129.
doi: 10.1016/j.physleta.2008.10.050. |
[9] |
A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, J. Phys. A, 35 (2002).
doi: 10.1088/0305-4470/35/32/201. |
[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. |
[11] |
A. Constantin and B. Kolev, $H^k$ metrics on the diffeomorphism group of the circle,, J. Nonlin. Math. Phys., 10 (2003), 424.
doi: 10.2991/jnmp.2003.10.4.1. |
[12] |
A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and perturbation theory (Rome, (1999), 23.
|
[13] |
D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. Math., 92 (1970), 102.
doi: 10.2307/1970699. |
[14] |
J. Escher, Non-metric two-component Euler equations on the circle,, Monatsh. Math., 167 (2012), 449.
doi: 10.1007/s00605-011-0323-3. |
[15] |
J. Escher, D. Henry, B. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity,, Ann. Mat. Pur. App., (2014).
doi: 10.1007/s10231-014-0461-z. |
[16] |
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. |
[17] |
J. Escher and B. Kolev, Geodesic completeness for Sobolev $H^s$-metrics on the diffeomorphism group of the circle,, J. Evol. Equ., 6 (2014), 335.
doi: 10.1007/s00028-014-0245-3. |
[18] |
J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle,, J. Geom. Mech., 14 (2014), 949.
doi: 10.3934/jgm.2014.6.335. |
[19] |
J. Escer and Z. Yin, Well-posedness, blow-up phenomena and global solutions for the $b$-equation,, J. Reine Angew. Math., 624 (2008), 51.
doi: 10.1515/CRELLE.2008.080. |
[20] |
D. Henry, Compactly supported solutions of a family of nonlinear differential equations,, Dyn. Contin. Discrete Impuls Syst. Ser. A Math. Anal., 15 (2008), 145.
|
[21] |
D. Henry, Infinite propagation speed for a two component Camassa-Holm equation,, Discrete Dyn. Syst. Ser. B, 12 (2009), 597.
doi: 10.3934/dcdsb.2009.12.597. |
[22] |
D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlin. Anal., 70 (2009), 1565.
doi: 10.1016/j.na.2008.02.104. |
[23] |
R. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case,, Wave Motion, 46 (2009), 389.
doi: 10.1016/j.wavemoti.2009.06.012. |
[24] |
R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63.
doi: 10.1017/S0022112001007224. |
[25] |
R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow,, Fluid Dynam. Res., 33 (2003), 97.
doi: 10.1016/S0169-5983(03)00036-4. |
[26] |
B. Kolev, Some geometric investigations on the Degasperis-Procesi shallow water equation,, Wave Motion, 46 (2009), 412.
doi: 10.1016/j.wavemoti.2009.06.005. |
[27] |
G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080.
doi: 10.1007/PL00012648. |
show all references
References:
[1] |
V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319.
doi: 10.5802/aif.233. |
[2] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.
doi: 10.1103/PhysRevLett.71.1661. |
[3] |
R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1.
doi: 10.1016/S0065-2156(08)70254-0. |
[4] |
A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321.
doi: 10.5802/aif.1757. |
[5] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, Volume 81 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), (2011).
doi: 10.1137/1.9781611971873. |
[6] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.
doi: 10.1007/BF02392586. |
[7] |
A. Constantin and J. Escher, Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475.
doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. |
[8] |
A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129.
doi: 10.1016/j.physleta.2008.10.050. |
[9] |
A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, J. Phys. A, 35 (2002).
doi: 10.1088/0305-4470/35/32/201. |
[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. |
[11] |
A. Constantin and B. Kolev, $H^k$ metrics on the diffeomorphism group of the circle,, J. Nonlin. Math. Phys., 10 (2003), 424.
doi: 10.2991/jnmp.2003.10.4.1. |
[12] |
A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and perturbation theory (Rome, (1999), 23.
|
[13] |
D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. Math., 92 (1970), 102.
doi: 10.2307/1970699. |
[14] |
J. Escher, Non-metric two-component Euler equations on the circle,, Monatsh. Math., 167 (2012), 449.
doi: 10.1007/s00605-011-0323-3. |
[15] |
J. Escher, D. Henry, B. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity,, Ann. Mat. Pur. App., (2014).
doi: 10.1007/s10231-014-0461-z. |
[16] |
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. |
[17] |
J. Escher and B. Kolev, Geodesic completeness for Sobolev $H^s$-metrics on the diffeomorphism group of the circle,, J. Evol. Equ., 6 (2014), 335.
doi: 10.1007/s00028-014-0245-3. |
[18] |
J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle,, J. Geom. Mech., 14 (2014), 949.
doi: 10.3934/jgm.2014.6.335. |
[19] |
J. Escer and Z. Yin, Well-posedness, blow-up phenomena and global solutions for the $b$-equation,, J. Reine Angew. Math., 624 (2008), 51.
doi: 10.1515/CRELLE.2008.080. |
[20] |
D. Henry, Compactly supported solutions of a family of nonlinear differential equations,, Dyn. Contin. Discrete Impuls Syst. Ser. A Math. Anal., 15 (2008), 145.
|
[21] |
D. Henry, Infinite propagation speed for a two component Camassa-Holm equation,, Discrete Dyn. Syst. Ser. B, 12 (2009), 597.
doi: 10.3934/dcdsb.2009.12.597. |
[22] |
D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlin. Anal., 70 (2009), 1565.
doi: 10.1016/j.na.2008.02.104. |
[23] |
R. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case,, Wave Motion, 46 (2009), 389.
doi: 10.1016/j.wavemoti.2009.06.012. |
[24] |
R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63.
doi: 10.1017/S0022112001007224. |
[25] |
R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow,, Fluid Dynam. Res., 33 (2003), 97.
doi: 10.1016/S0169-5983(03)00036-4. |
[26] |
B. Kolev, Some geometric investigations on the Degasperis-Procesi shallow water equation,, Wave Motion, 46 (2009), 412.
doi: 10.1016/j.wavemoti.2009.06.005. |
[27] |
G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080.
doi: 10.1007/PL00012648. |
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