Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 027, 14 pages      arXiv:1211.6931      https://doi.org/10.3842/SIGMA.2013.027
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

G-Strands and Peakon Collisions on Diff(R)

Darryl D. Holm a and Rossen I. Ivanov b
a) Department of Mathematics, Imperial College London, London SW7 2AZ, UK
b) School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

Received October 29, 2012, in final form March 21, 2013; Published online March 26, 2013

Abstract
A G-strand is a map g: R×R→G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that G-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of G-strands when G=Diff(R) is the group of diffeomorphisms of the real line R, for which the group product is composition of smooth invertible functions. In the case of peakon-antipeakon collisions, the solution reduces to solving either Laplace's equation or the wave equation (depending on a sign in the Lagrangian) and is written in terms of their solutions. We also consider the complexified systems of G-strand equations for G=Diff(R) corresponding to a harmonic map g: C→Diff(R) and find explicit expressions for its peakon-antipeakon solutions, as well.

Key words: Hamilton's principle; continuum spin chains; Euler-Poincaré equations; Sobolev norms; singular momentum maps; diffeomorphisms; harmonic maps.

pdf (399 kb)   tex (22 kb)

References

  1. Camassa R., Holm D.D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664, patt-sol/9305002.
  2. Cherednik I.V., On the integrability of the 2-dimensional asymmetric chiral O(3) field equations and their quantum analogue, Soviet J. Nuclear Phys. 33 (1981), 144-145.
  3. De Vega H.J., Field theories with an infinite number of conservation laws and Bäcklund transformations in two dimensions, Phys. Lett. B 87 (1979), 233-236.
  4. Ellis D.C.P., Gay-Balmaz F., Holm D.D., Putkaradze V., Ratiu T.S., Symmetry reduced dynamics of charged molecular strands, Arch. Ration. Mech. Anal. 197 (2010), 811-902, arXiv:0901.2959.
  5. Gay-Balmaz F., Clebsch variational principles in field theories and singular solutions of covariant EPDiff equations, Rep. Math. Phys. 71 (2013), 231-277, arXiv:1209.0109.
  6. Gay-Balmaz F., Ratiu T.S., The geometric structure of complex fluids, Adv. in Appl. Math. 42 (2009), 176-275, arXiv:0903.4294.
  7. Gibbons J., Holm D.D., Kupershmidt B., Gauge-invariant Poisson brackets for chromohydrodynamics, Phys. Lett. A 90 (1982), 281-283.
  8. Gibbons J., Holm D.D., Kupershmidt B., The Hamiltonian structure of classical chromohydrodynamics, Phys. D 6 (1983), 179-194.
  9. Guest M.A., Harmonic maps, loop groups, and integrable systems, London Mathematical Society Student Texts, Vol. 38, Cambridge University Press, Cambridge, 1997.
  10. Guha P., Olver P.J., Geodesic flow and two (super) component analog of the Camassa-Holm equation, SIGMA 2 (2006), 054, 9 pages, nlin.SI/0605041.
  11. Hlavatý L., Snobl L., Principal chiral models on non-semisimple groups, J. Phys. A: Math. Gen. 34 (2001), 7795-7809.
  12. Holm D.D., Euler-Poincaré dynamics of perfect complex fluids, in Geometry, Mechanics, and Dynamics, Editors P. Newton, P. Holmes, A. Weinstein, Springer, New York, 2002, 113-167, nlin.CD/0103041.
  13. Holm D.D., Geometric mechanics. Part II. Rotating, translating and rolling, 2nd ed., Imperial College Press, London, 2011.
  14. Holm D.D., Ivanov R.I., Percival J.R., G-strands, J. Nonlinear Sci. 22 (2012), 517-551, arXiv:1109.4421.
  15. Holm D.D., Kupershmidt B.A., The analogy between spin glasses and Yang-Mills fluids, J. Math. Phys. 29 (1988), 21-30.
  16. Holm D.D., Marsden J.E., Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation, in The Breadth of Symplectic and Poisson Geometry, Progr. Math., Vol. 232, Editors J.E. Marsden, T.S. Ratiu, Birkhäuser Boston, Boston, MA, 2005, 203-235, nlin.CD/0312048.
  17. Holm D.D., Marsden J.E., Ratiu T.S., The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1998), 1-81, chao-dyn/9801015.
  18. Lax P.D., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490.
  19. Maillet J.-M., Hamiltonian structures for integrable classical theories from graded Kac-Moody algebras, Phys. Lett. B 167 (1986), 401-405.
  20. Mañas M., The principal chiral model as an integrable system, in Harmonic Maps and Integrable Systems, Aspects Math., Vol. E23, Vieweg, Braunschweig, 1994, 147-173.
  21. Marsden J.E., Ratiu T.S., Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems, Texts in Applied Mathematics, Vol. 17, Springer-Verlag, New York, 1994.
  22. Novikov S., Manakov S.V., Pitaevski L.P., Zakharov V.E., Theory of solitons. The inverse scattering method, Contemporary Soviet Mathematics, Plenum, New York, 1984.
  23. Pohlmeyer K., Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976), 207-221.
  24. Popowicz Z., A 2-component or N=2 supersymmetric Camassa-Holm equation, Phys. Lett. A 354 (2006), 110-114, nlin.SI/0509050.
  25. Sochen N., Integrable generalized principal chiral models, Phys. Lett. B 391 (1997), 374-380, hep-th/9607009.
  26. Tronci C., Hybrid models for complex fluids with multipolar interactions, J. Geom. Mech. 4 (2012), 333-363, arXiv:1011.4389.
  27. Uhlenbeck K., Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom. 30 (1989), 1-50.
  28. Ward R.S., Soliton solutions in an integrable chiral model in 2+1 dimensions, J. Math. Phys. 29 (1988), 386-389.
  29. Witten E., Non-abelian bosonization in two dimensions, Comm. Math. Phys. 92 (1984), 455-472.
  30. Yanovski A.B., Bi-Hamiltonian formulation of the O(3) chiral fields equations hierarchy via a polynomial bundle, J. Phys. A: Math. Gen. 31 (1998), 8709-8726.
  31. Zakharov V.E., Mikhailov A.V., On the integrability of classical spinor models in two-dimensional space-time, Comm. Math. Phys. 74 (1980), 21-40.
  32. Zakharov V.E., Mikhailov A.V., Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Soviet Phys. JETP 47 (1978), 1017-1027.

Previous article  Next article   Contents of Volume 9 (2013)