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Discriminantly separable polynomials and quad-equations
Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
1. | Institute for Applied Mathematics, University of Hanover, D-30167 Hanover |
2. | Institut de Mathématiques de Marseille, Aix Marseille Université, CNRS, Centrale Marseille, 13453 Marseille, France |
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] |
M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group,, Ann. Global Anal. Geom., 44 (2013), 5.
doi: 10.1007/s10455-012-9353-x. |
[3] |
A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal., 37 (2005), 996.
doi: 10.1137/050623036. |
[4] |
M. Bruveris, The energy functional on the Virasoro-Bott group with the $L^{2}$-metric has no local minima,, Ann. Global Anal. Geom., 43 (2013), 385.
doi: 10.1007/s10455-012-9350-0. |
[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. |
[6] |
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. |
[7] |
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. |
[8] |
P. Constantin, P. D. Lax and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation,, Comm. Pure Appl. Math., 38 (1985), 715.
doi: 10.1002/cpa.3160380605. |
[9] |
D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math. (2), 92 (1970), 102.
doi: 10.2307/1970699. |
[10] |
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. |
[11] |
J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation,, Commun. Pure Appl. Anal., 11 (2012), 1407.
doi: 10.3934/cpaa.2012.11.1407. |
[12] |
J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, Communications in Contemporary Mathematics, 14 (2012).
doi: 10.1142/S0219199712500162. |
[13] |
L. P. Euler, Du mouvement de rotation des corps solides autour d'un axe variable,, Mémoires de l'académie des sciences de Berlin, 14 (1765), 154. Google Scholar |
[14] |
A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47.
doi: 10.1016/0167-2789(81)90004-X. |
[15] |
F. Gay-Balmaz, Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space,, PhD thesis, (2009). Google Scholar |
[16] |
E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian geometry on infinite-dimensional manifolds,, ArXiv e-prints, ().
doi: 10.1007/s12220-014-9523-0. |
[17] |
E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian structures corresponding to Kählerian metrics on the universal Teichmüller space and curve,, ArXiv e-prints, (). Google Scholar |
[18] |
L. Guieu and C. Roger, L'algèbre et le Groupe de Virasoro,, Les Publications CRM, (2007).
|
[19] |
R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65.
doi: 10.1090/S0273-0979-1982-15004-2. |
[20] |
D. D. Holm, J. E. 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. |
[21] |
J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498.
doi: 10.1137/0151075. |
[22] |
H. Inci, T. Kappeler and P. Topalov, On the Regularity of the Composition of Diffeomorphisms, vol. 226 of Memoirs of the American Mathematical Society,, 1st edition, (2013).
doi: 10.1090/S0065-9266-2013-00676-4. |
[23] |
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.
doi: 10.1007/s00208-008-0250-3. |
[24] |
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. |
[25] |
B. Kolev, Lie groups and mechanics: An introduction,, J. Nonlinear Math. Phys., 11 (2004), 480.
doi: 10.2991/jnmp.2004.11.4.5. |
[26] |
S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics,, Springer-Verlag, (1999). Google Scholar |
[27] |
J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere,, J. Geom. Phys., 57 (2007), 2049.
doi: 10.1016/j.geomphys.2007.05.003. |
[28] |
J. Lenells, G. Misiołek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions,, Comm. Math. Phys., 299 (2010), 129.
doi: 10.1007/s00220-010-1069-9. |
[29] |
P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217.
|
[30] |
J. Milnor, Remarks on infinite-dimensional Lie groups,, in Relativity, (1984), 1007.
|
[31] |
G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203.
doi: 10.1016/S0393-0440(97)00010-7. |
[32] |
G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080.
doi: 10.1007/PL00012648. |
[33] |
S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle,, Osaka J. Math., 32 (1995), 1.
|
[34] |
P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics,, 2nd edition, (1993).
doi: 10.1007/978-1-4612-4350-2. |
[35] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[36] |
H. Poincaré, Sur une nouvelle forme des équations de la mécanique,, C.R. Acad. Sci., 132 (1901), 369. Google Scholar |
[37] |
S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics,, J. Funct. Anal., 160 (1998), 337.
doi: 10.1006/jfan.1998.3335. |
[38] |
L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space,, Mem. Amer. Math. Soc., 183 (2006).
doi: 10.1090/memo/0861. |
[39] |
F. Tiǧlay and C. Vizman, Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications,, Lett. Math. Phys., 97 (2011), 45.
doi: 10.1007/s11005-011-0464-2. |
[40] |
H. Triebel, Theory of Function Spaces,, Birkhäuser Boston, (1983).
doi: 10.1007/978-3-0346-0416-1. |
[41] |
M. Wunsch, On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric,, J. Nonlinear Math. Phys., 17 (2010), 7.
doi: 10.1142/S1402925110000544. |
[42] |
Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272.
doi: 10.1137/S0036141003425672. |
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] |
M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group,, Ann. Global Anal. Geom., 44 (2013), 5.
doi: 10.1007/s10455-012-9353-x. |
[3] |
A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal., 37 (2005), 996.
doi: 10.1137/050623036. |
[4] |
M. Bruveris, The energy functional on the Virasoro-Bott group with the $L^{2}$-metric has no local minima,, Ann. Global Anal. Geom., 43 (2013), 385.
doi: 10.1007/s10455-012-9350-0. |
[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. |
[6] |
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. |
[7] |
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. |
[8] |
P. Constantin, P. D. Lax and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation,, Comm. Pure Appl. Math., 38 (1985), 715.
doi: 10.1002/cpa.3160380605. |
[9] |
D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math. (2), 92 (1970), 102.
doi: 10.2307/1970699. |
[10] |
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. |
[11] |
J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation,, Commun. Pure Appl. Anal., 11 (2012), 1407.
doi: 10.3934/cpaa.2012.11.1407. |
[12] |
J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, Communications in Contemporary Mathematics, 14 (2012).
doi: 10.1142/S0219199712500162. |
[13] |
L. P. Euler, Du mouvement de rotation des corps solides autour d'un axe variable,, Mémoires de l'académie des sciences de Berlin, 14 (1765), 154. Google Scholar |
[14] |
A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47.
doi: 10.1016/0167-2789(81)90004-X. |
[15] |
F. Gay-Balmaz, Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space,, PhD thesis, (2009). Google Scholar |
[16] |
E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian geometry on infinite-dimensional manifolds,, ArXiv e-prints, ().
doi: 10.1007/s12220-014-9523-0. |
[17] |
E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian structures corresponding to Kählerian metrics on the universal Teichmüller space and curve,, ArXiv e-prints, (). Google Scholar |
[18] |
L. Guieu and C. Roger, L'algèbre et le Groupe de Virasoro,, Les Publications CRM, (2007).
|
[19] |
R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65.
doi: 10.1090/S0273-0979-1982-15004-2. |
[20] |
D. D. Holm, J. E. 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. |
[21] |
J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498.
doi: 10.1137/0151075. |
[22] |
H. Inci, T. Kappeler and P. Topalov, On the Regularity of the Composition of Diffeomorphisms, vol. 226 of Memoirs of the American Mathematical Society,, 1st edition, (2013).
doi: 10.1090/S0065-9266-2013-00676-4. |
[23] |
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.
doi: 10.1007/s00208-008-0250-3. |
[24] |
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. |
[25] |
B. Kolev, Lie groups and mechanics: An introduction,, J. Nonlinear Math. Phys., 11 (2004), 480.
doi: 10.2991/jnmp.2004.11.4.5. |
[26] |
S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics,, Springer-Verlag, (1999). Google Scholar |
[27] |
J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere,, J. Geom. Phys., 57 (2007), 2049.
doi: 10.1016/j.geomphys.2007.05.003. |
[28] |
J. Lenells, G. Misiołek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions,, Comm. Math. Phys., 299 (2010), 129.
doi: 10.1007/s00220-010-1069-9. |
[29] |
P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217.
|
[30] |
J. Milnor, Remarks on infinite-dimensional Lie groups,, in Relativity, (1984), 1007.
|
[31] |
G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203.
doi: 10.1016/S0393-0440(97)00010-7. |
[32] |
G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080.
doi: 10.1007/PL00012648. |
[33] |
S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle,, Osaka J. Math., 32 (1995), 1.
|
[34] |
P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics,, 2nd edition, (1993).
doi: 10.1007/978-1-4612-4350-2. |
[35] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[36] |
H. Poincaré, Sur une nouvelle forme des équations de la mécanique,, C.R. Acad. Sci., 132 (1901), 369. Google Scholar |
[37] |
S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics,, J. Funct. Anal., 160 (1998), 337.
doi: 10.1006/jfan.1998.3335. |
[38] |
L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space,, Mem. Amer. Math. Soc., 183 (2006).
doi: 10.1090/memo/0861. |
[39] |
F. Tiǧlay and C. Vizman, Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications,, Lett. Math. Phys., 97 (2011), 45.
doi: 10.1007/s11005-011-0464-2. |
[40] |
H. Triebel, Theory of Function Spaces,, Birkhäuser Boston, (1983).
doi: 10.1007/978-3-0346-0416-1. |
[41] |
M. Wunsch, On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric,, J. Nonlinear Math. Phys., 17 (2010), 7.
doi: 10.1142/S1402925110000544. |
[42] |
Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272.
doi: 10.1137/S0036141003425672. |
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