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

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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

Abstract
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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.
[2]	V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.
[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.
[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.
[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.
[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.
[7]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.
[28]	I. Vaisman, "Lectures on the Geometry of Poisson Manifolds," Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994.
[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.

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.
[2]	V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.
[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.
[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.
[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.
[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.
[7]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.
[28]	I. Vaisman, "Lectures on the Geometry of Poisson Manifolds," Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994.
[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.

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