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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), 319-361. doi: 10.5802/aif.233.  Google Scholar

[2]

V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singapore), 5 (2007), 1-27. 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), 215-239. 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), 1661-1664. 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), 321-362. 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), 229-243. 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), 75-91. 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), 7129-7132. 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), R51-R79. 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), 787-804. 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), 109-122. 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), 165-186. 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), 265-205. 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, published online, 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), 92-93.  Google Scholar

[17]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222.  Google Scholar

[18]

D. J. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347. 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), 13-49. 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 "The Breadth of Symplectic and Poisson Geometry" (eds. J. E. Marsden and T. S. Ratiu), Progr. Math., 232, Birkhäuser Boston, Boston, MA, (2005), 203-235.  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), 1-81. 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), 2267-2280. 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), 389-396. 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), 63-82. doi: 10.1017/S0022112001007224.  Google Scholar

[25]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144. 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), 2333-2357.  Google Scholar

[27]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.  Google Scholar

[28]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds," Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 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), 319-361. doi: 10.5802/aif.233.  Google Scholar

[2]

V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singapore), 5 (2007), 1-27. 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), 215-239. 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), 1661-1664. 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), 321-362. 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), 229-243. 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), 75-91. 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), 7129-7132. 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), R51-R79. 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), 787-804. 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), 109-122. 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), 165-186. 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), 265-205. 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, published online, 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), 92-93.  Google Scholar

[17]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222.  Google Scholar

[18]

D. J. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347. 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), 13-49. 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 "The Breadth of Symplectic and Poisson Geometry" (eds. J. E. Marsden and T. S. Ratiu), Progr. Math., 232, Birkhäuser Boston, Boston, MA, (2005), 203-235.  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), 1-81. 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), 2267-2280. 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), 389-396. 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), 63-82. doi: 10.1017/S0022112001007224.  Google Scholar

[25]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144. 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), 2333-2357.  Google Scholar

[27]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.  Google Scholar

[28]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds," Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 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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