Moving frames, Geometric Poisson brackets and the KdV-Schwarzian evolution of pure spinors

Gloria Marí Beffa[1]

  • [1] University of Wisconsin Mathematics department Madison, Wisconsin 53706 (USA)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 6, page 2405-2434
  • ISSN: 0373-0956

Abstract

top
In this paper we describe a non-local moving frame along a curve of pure spinors in O ( 2 m , 2 m ) / P , and its associated basis of differential invariants. We show that the space of differential invariants of Schwarzian-type define a Poisson submanifold of the spinor Geometric Poisson brackets. The resulting restriction is given by a decoupled system of KdV Poisson structures. We define a generalization of the Schwarzian-KdV evolution for pure spinor curves and we prove that it induces a decoupled system of KdV equations on the invariants of projective type, when restricted to a certain level set. We also describe its associated Miura transformation and non-commutative modified KdV system.

How to cite

top

Marí Beffa, Gloria. "Moving frames, Geometric Poisson brackets and the KdV-Schwarzian evolution of pure spinors." Annales de l’institut Fourier 61.6 (2011): 2405-2434. <http://eudml.org/doc/219747>.

@article{MaríBeffa2011,
abstract = {In this paper we describe a non-local moving frame along a curve of pure spinors in $\mathrm\{O\}(2m,2m)/P$, and its associated basis of differential invariants. We show that the space of differential invariants of Schwarzian-type define a Poisson submanifold of the spinor Geometric Poisson brackets. The resulting restriction is given by a decoupled system of KdV Poisson structures. We define a generalization of the Schwarzian-KdV evolution for pure spinor curves and we prove that it induces a decoupled system of KdV equations on the invariants of projective type, when restricted to a certain level set. We also describe its associated Miura transformation and non-commutative modified KdV system.},
affiliation = {University of Wisconsin Mathematics department Madison, Wisconsin 53706 (USA)},
author = {Marí Beffa, Gloria},
journal = {Annales de l’institut Fourier},
keywords = {Moving frame; spinor evolutions; geometric Poisson brackets; KdV equations; differential invariants; Miura transformation; non-commutative modified KdV system; moving frame},
language = {eng},
number = {6},
pages = {2405-2434},
publisher = {Association des Annales de l’institut Fourier},
title = {Moving frames, Geometric Poisson brackets and the KdV-Schwarzian evolution of pure spinors},
url = {http://eudml.org/doc/219747},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Marí Beffa, Gloria
TI - Moving frames, Geometric Poisson brackets and the KdV-Schwarzian evolution of pure spinors
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 6
SP - 2405
EP - 2434
AB - In this paper we describe a non-local moving frame along a curve of pure spinors in $\mathrm{O}(2m,2m)/P$, and its associated basis of differential invariants. We show that the space of differential invariants of Schwarzian-type define a Poisson submanifold of the spinor Geometric Poisson brackets. The resulting restriction is given by a decoupled system of KdV Poisson structures. We define a generalization of the Schwarzian-KdV evolution for pure spinor curves and we prove that it induces a decoupled system of KdV equations on the invariants of projective type, when restricted to a certain level set. We also describe its associated Miura transformation and non-commutative modified KdV system.
LA - eng
KW - Moving frame; spinor evolutions; geometric Poisson brackets; KdV equations; differential invariants; Miura transformation; non-commutative modified KdV system; moving frame
UR - http://eudml.org/doc/219747
ER -

References

top
  1. Stephen C. Anco, Bi-Hamiltonian operators, integrable flows of curves using moving frames and geometric map equations, J. Phys. A 39 (2006), 2043-2072 Zbl1085.37049MR2211976
  2. Stephen C. Anco, Hamiltonian flows of curves in G / SO ( N ) and vector soliton equations of mKdV and sine-Gordon type, SIGMA Symmetry Integrability Geom. Methods Appl. 2 (2006) Zbl1102.37042MR2217753
  3. Toby N. Bailey, Michael G. Eastwood, Conformal circles and parametrizations of curves in conformal manifolds, Proc. Amer. Math. Soc. 108 (1990), 215-221 Zbl0684.53016
  4. Toby N. Bailey, Michael G. Eastwood, Complex paraconformal manifolds—their differential geometry and twistor theory, Forum Math. 3 (1991), 61-103 Zbl0728.53005MR1085595
  5. Annalisa Calini, Thomas Ivey, Gloria Marí-Beffa, Remarks on KdV-type flows on star-shaped curves, Phys. D 238 (2009), 788-797 Zbl1218.37102MR2522973
  6. Élie Cartan, La Méthode du Repère Mobile, la Théorie des Groupes Continus et les Espaces Généralisés, Exposés de Géométrie 5 (1935), Hermann, Paris Zbl0010.39501
  7. Élie Cartan, Les espaces à connexion conforme, Oeuvres Complètes III.1 (1955), 747-797, Gauthier-Villars, Paris 
  8. Kai-Seng Chou, Changzheng Qu, Integrable equations arising from motions of plane curves, Phys. D 162 (2002), 9-33 Zbl0987.35139MR1882237
  9. Kai-Seng Chou, Changzheng Qu, Integrable equations arising from motions of plane curves. II, J. Nonlinear Sci. 13 (2003), 487-517 Zbl1045.35063MR1882237
  10. V. G. Drinfeld, V. V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, Current problems in mathematics, Vol. 24 (1984), 81-180, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow Zbl0558.58027MR760998
  11. B. A. Dubrovin, S. P. Novikov, Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Uspekhi Mat. Nauk 44 (1989), 29-98, 203 Zbl0712.58032MR1037010
  12. M. Eastwood, G. Marí Beffa, Geometric Poisson brackets on Grassmannian and conformal spheres Zbl1248.53068
  13. Mark Fels, Peter J. Olver, Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213 Zbl0937.53012MR1620769
  14. Mark Fels, Peter J. Olver, Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208 Zbl0937.53013MR1681815
  15. A. Fialkov, The Conformal Theory of Curves, Transactions of the AMS 51 (1942), 435-568 Zbl0063.01358MR6465
  16. François Gay-Balmaz, Tudor S. Ratiu, Group actions on chains of Banach manifolds and applications to fluid dynamics, Ann. Global Anal. Geom. 31 (2007), 287-328 Zbl1122.58006MR2314804
  17. Mark L. Green, The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces, Duke Math. J. 45 (1978), 735-779 Zbl0414.53039MR518104
  18. P. Griffiths, On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 775-814 Zbl0294.53034MR410607
  19. R. Hasimoto, A soliton on a vortex filament, J. Fluid Mechanics 51 (1972), 477-485 Zbl0237.76010
  20. N. J. Hitchin, G. B. Segal, R. S. Ward, Integrable systems, 4 (1999), The Clarendon Press Oxford University Press, New York Zbl1082.37501MR1723384
  21. Rongpei Huang, David A. Singer, A new flow on starlike curves in ℝ 3 , Proc. Amer. Math. Soc. 130 (2002), 2725-2735 (electronic) Zbl1007.53007MR1900890
  22. E. Hubert, Generation properties of differential invariants in the moving frame methods 
  23. A. A. Kirillov, Infinite-dimensional groups, their representations, orbits, invariants, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) (1980), 705-708, Acad. Sci. Fennica, Helsinki Zbl0427.22012MR562675
  24. Joel Langer, Ron Perline, Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991), 71-93 Zbl0795.35115MR1102831
  25. Joel Langer, Ron Perline, Geometric realizations of Fordy-Kulish nonlinear Schrödinger systems, Pacific J. Math. 195 (2000), 157-178 Zbl1115.37353MR1781618
  26. Elizabeth L. Mansfield, Peter H. van der Kamp, Evolution of curvature invariants and lifting integrability, J. Geom. Phys. 56 (2006), 1294-1325 Zbl1099.53012MR2236264
  27. G. Marí Beffa, P. J. Olver, Poisson structures for geometric curve flows in semi-simple homogeneous spaces, Regul. Chaotic Dyn. 15 (2010), 532-550 Zbl1229.22018MR2679763
  28. G. Marí Beffa, J. A. Sanders, Jing Ping Wang, Integrable systems in three-dimensional Riemannian geometry, J. Nonlinear Sci. 12 (2002), 143-167 Zbl1140.37361MR1894465
  29. Gloria Marí Beffa, On completely integrable geometric evolutions of curves of Lagrangian planes, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 111-131 Zbl1130.37032MR2359775
  30. Gloria Marí Beffa, Geometric Poisson brackets in flat semisimple homogenous spaces, Asian Journal of Mathematics 12 (2008), 1-33 Zbl1173.37054MR2415008
  31. Gloria Marí Beffa, Geometric realizations of bi-Hamiltonian completely integrable systems, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008) Zbl1157.37336MR2393293
  32. Gloria Marí Beffa, Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds, Ann. Inst. Fourier (Grenoble) 58 (2008), 1295-1335 Zbl1192.37099MR2427961
  33. Gloria Marí Beffa, On bi-Hamiltonian flows and their realizations as curves in real homogeneous manifold, Pacific Journal of Mathematics 247 (2010), 163-188 Zbl1213.37096MR2718210
  34. Peter J. Olver, Equivalence, invariants, and symmetry, (1995), Cambridge University Press, Cambridge Zbl0837.58001MR1337276
  35. Valentin Ovsienko, Lagrange Schwarzian derivative and symplectic Sturm theory, Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), 73-96 Zbl0780.34004MR1230706
  36. Valentin Ovsienko, S. Tabachnikov, Projective differential geometry old and new, 165 (2005), Cambridge University Press, Cambridge Zbl1073.53001MR2177471
  37. Jan A. Sanders, Jing Ping Wang, Integrable systems in n -dimensional Riemannian geometry, Mosc. Math. J. 3 (2003), 1369-1393 Zbl1050.37035MR2058803
  38. Graeme Segal, The geometry of the KdV equation, Internat. J. Modern Phys. A 6 (1991), 2859-2869 Zbl0741.35073MR1117753
  39. Shane A. Squires, Gloria Marí Beffa, Integrable systems associated to curves in flat Galilean and Minkowski spaces, Appl. Anal. 89 (2010), 571-592 Zbl1218.37095MR2647767
  40. Chuu-Lian Terng, Gudlaugur Thorbergsson, Completely integrable curve flows on adjoint orbits, Results Math. 40 (2001), 286-309 Zbl1023.37041MR1860376
  41. Chuu-Lian Terng, Karen Uhlenbeck, Schrödinger flows on Grassmannians, Integrable systems, geometry, and topology 36 (2006), 235-256, Amer. Math. Soc., Providence, RI Zbl1110.37056MR2222517

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.