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September  2011, 3(3): 313-322. doi: 10.3934/jgm.2011.3.313

Euler equations on a semi-direct product of the diffeomorphisms group by itself

Joachim Escher 1, , Rossen Ivanov 2, and  Boris Kolev 3,

1. 

Institute for Applied Mathematics, University of Hanover, D-30167 Hanover, Germany
2. 

School of Mathematical Science, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
3. 

LATP, CNRS & University of Provence, 39 Rue F. Joliot-Curie, 13453 Marseille Cedex 13

Received  August 2011 Revised  September 2011 Published  November 2011

The geodesic equations of a class of right invariant metrics on the semi-direct product $Diff(\mathbb{S}^1)$Ⓢ$Diff(\mathbb{S}^1)$ are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra $(Vect(\mathbb{S}^1)$Ⓢ$Vect(\mathbb{S}^1))^{*}$ are found.
Keywords: diffeomorphism group of the circle., peakons, integrable systems, Euler equation.
Mathematics Subject Classification: Primary: 58D05, 37K65, 58B20; Secondary: 35Q5.
Citation: Joachim Escher, Rossen Ivanov, Boris Kolev. Euler equations on a semi-direct product of the diffeomorphisms group by itself. Journal of Geometric Mechanics, 2011, 3 (3) : 313-322. doi: 10.3934/jgm.2011.3.313
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.  Google Scholar

[2]

V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics,", Applied Mathematical Sciences, 125 (1998).   Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl. (Singapore), 5 (2007), 1.  doi: 10.1142/S0219530507000857.  Google Scholar

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215.  doi: 10.1007/s00205-006-0010-z.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.  doi: 10.1007/BF02392586.  Google Scholar

[8]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75.  doi: 10.1007/PL00004793.  Google Scholar

[9]

A. Constantin and R. 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.  Google Scholar

[10]

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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

C. Cotter, D. Holm, R. I. Ivanov and J. R. Percival, Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation,, J. Phys. A: Math. Theor., 44 (2011), 265.  doi: 10.1088/1751-8113/44/26/265205.  Google Scholar

[15]

J. Escher, Non-metric two-component Euler equations on the circle,, Monatshefte für Mathematik, (2011).  doi: 10.1007/s00605-011-0323-3.  Google Scholar

[16]

I. M. Gel'fand and D. B. Fuks, Cohomologies of the Lie algebra of vector fields on the circle,, Funkcional. Anal. i Priložen, 2 (1968), 92.   Google Scholar

[17]

R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65.   Google Scholar

[18]

D. J. Henry, Compactly supported solutions of the Camassa-Holm equation,, J. Nonlinear Math. Phys., 12 (2005), 342.  doi: 10.2991/jnmp.2005.12.3.3.  Google Scholar

[19]

D. D. Holm and R. I. Ivanov, Smooth and peaked solitons of the Camassa-Holm equation and applications,, J. of Geometry and Symmetry in Physics, 22 (2011), 13.   Google Scholar

[20]

D. D. Holm and J. E.Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation,, in, 232 (2005), 203.   Google Scholar

[21]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar

[22]

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

[23]

R. I. 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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2333.   Google Scholar

[27]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203.   Google Scholar

[28]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,", Progress in Mathematics, 118 (1994).   Google Scholar

[29]

P. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,, Int. Math. Res. Not. IMRN, 2010 (): 1981.   Google Scholar

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.  Google Scholar

[2]

V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics,", Applied Mathematical Sciences, 125 (1998).   Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl. (Singapore), 5 (2007), 1.  doi: 10.1142/S0219530507000857.  Google Scholar

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215.  doi: 10.1007/s00205-006-0010-z.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.  doi: 10.1007/BF02392586.  Google Scholar

[8]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75.  doi: 10.1007/PL00004793.  Google Scholar

[9]

A. Constantin and R. 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.  Google Scholar

[10]

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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

C. Cotter, D. Holm, R. I. Ivanov and J. R. Percival, Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation,, J. Phys. A: Math. Theor., 44 (2011), 265.  doi: 10.1088/1751-8113/44/26/265205.  Google Scholar

[15]

J. Escher, Non-metric two-component Euler equations on the circle,, Monatshefte für Mathematik, (2011).  doi: 10.1007/s00605-011-0323-3.  Google Scholar

[16]

I. M. Gel'fand and D. B. Fuks, Cohomologies of the Lie algebra of vector fields on the circle,, Funkcional. Anal. i Priložen, 2 (1968), 92.   Google Scholar

[17]

R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65.   Google Scholar

[18]

D. J. Henry, Compactly supported solutions of the Camassa-Holm equation,, J. Nonlinear Math. Phys., 12 (2005), 342.  doi: 10.2991/jnmp.2005.12.3.3.  Google Scholar

[19]

D. D. Holm and R. I. Ivanov, Smooth and peaked solitons of the Camassa-Holm equation and applications,, J. of Geometry and Symmetry in Physics, 22 (2011), 13.   Google Scholar

[20]

D. D. Holm and J. E.Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation,, in, 232 (2005), 203.   Google Scholar

[21]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar

[22]

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

[23]

R. I. 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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2333.   Google Scholar

[27]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203.   Google Scholar

[28]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,", Progress in Mathematics, 118 (1994).   Google Scholar

[29]

P. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,, Int. Math. Res. Not. IMRN, 2010 (): 1981.   Google Scholar

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